Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Law of large numbers wikipedia, lookup

Hyperreal number wikipedia, lookup

Collatz conjecture wikipedia, lookup

Proofs of Fermat's little theorem wikipedia, lookup

Elementary mathematics wikipedia, lookup

Large numbers wikipedia, lookup

Elementary arithmetic wikipedia, lookup

Mathematics of radio engineering wikipedia, lookup

Transcript

Chapter 10 Resource Masters StudentWorks PlusTM includes the entire Student Edition text along with the worksheets in this booklet. TeacherWorks PlusTM includes all of the materials found in this booklet for viewing, printing, and editing. Cover: Jason Reed/Photodisc/Getty Images Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Permission is granted to reproduce the material contained herein on the condition that such materials be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with the Glencoe Precalculus program. Any other reproduction, for sale or other use, is expressly prohibited. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240 - 4027 ISBN: 978-0-07-893811-5 MHID: 0-07-893811-2 Printed in the United States of America. 2 3 4 5 6 7 8 9 10 079 18 17 16 15 14 13 12 11 10 Contents Teacher’s Guide to Using the Chapter 10 Resource Masters ........................................... iv Lesson 10-4 Mathematical Induction Study Guide and Intervention .......................... 21 Practice............................................................ 23 Word Problem Practice ................................... 24 Enrichment ...................................................... 25 Chapter Resources Student-Built Glossary ....................................... 1 Anticipation Guide (English) .............................. 3 Anticipation Guide (Spanish) ............................. 4 Lesson 10-5 Lesson 10-1 The Binomial Theorem Study Guide and Intervention .......................... 26 Practice............................................................ 28 Word Problem Practice ................................... 29 Enrichment ...................................................... 30 Sequences, Series, and Sigma Notation Study Guide and Intervention ............................ 5 Practice.............................................................. 7 Word Problem Practice ..................................... 8 Enrichment ........................................................ 9 Lesson 10-6 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Lesson 10-2 Arithmetic Sequences and Series Study Guide and Intervention .......................... 10 Practice............................................................ 12 Word Problem Practice ................................... 13 Enrichment ...................................................... 14 Representing Functions as Infinite Series Study Guide and Intervention .......................... 31 Practice............................................................ 33 Word Problem Practice ................................... 34 Enrichment ...................................................... 35 Graphing Calculator Activity ............................ 36 Lesson 10-3 Assessment Geometric Sequences and Series Study Guide and Intervention .......................... 15 Practice............................................................ 17 Word Problem Practice ................................... 18 Enrichment ...................................................... 19 TI-Nspire Activity ............................................. 20 Chapter 10 Quizzes 1 and 2 ........................... 37 Chapter 10 Quizzes 3 and 4 ........................... 38 Chapter 10 Mid-Chapter Test .......................... 39 Chapter 10 Vocabulary Test ........................... 40 Chapter 10 Test, Form 1 ................................. 41 Chapter 10 Test, Form 2A............................... 43 Chapter 10 Test, Form 2B............................... 45 Chapter 10 Test, Form 2C .............................. 47 Chapter 10 Test, Form 2D .............................. 49 Chapter 10 Test, Form 3 ................................. 51 Chapter 10 Extended-Response Test ............. 53 Standardized Test Practice ............................. 54 Answers ........................................... A1–A26 Chapter 10 iii Glencoe Precalculus Teacher’s Guide to Using the Chapter 10 Resource Masters The Chapter 10 Resource Masters includes the core materials needed for Chapter 10. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet. Practice This master closely follows the types of problems found in the Exercises section of the Student Edition and includes word problems. Use as an additional practice option or as homework for second-day teaching of the lesson. Chapter Resources Student-Built Glossary (pages 1–2) These masters are a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to record definitions and/or examples for each term. You may suggest that students highlight or star the terms with which they are not familiar. Give this to students before beginning Lesson 10-1. Encourage them to add these pages to their mathematics study notebooks. Remind them to complete the appropriate words as they study each lesson. Word Problem Practice This master includes additional practice in solving word problems that apply to the concepts of the lesson. Use as an additional practice or as homework for second-day teaching of the lesson. Enrichment These activities may extend the concepts of the lesson, offer an historical or multicultural look at the concepts, or widen students’ perspectives on the mathematics they are learning. They are written for use with all levels of students. Graphing Calculator, TI–Nspire, or Spreadsheet Activities These activities present ways in which technology can be used with the concepts in some lessons of this chapter. Use as an alternative approach to some concepts or as an integral part of your lesson presentation. Lesson Resources Study Guide and Intervention These masters provide vocabulary, key concepts, additional worked-out examples and Guided Practice exercises to use as a reteaching activity. It can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent. Chapter 10 iv Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Anticipation Guide (pages 3–4) This master, presented in both English and Spanish, is a survey used before beginning the chapter to pinpoint what students may or may not know about the concepts in the chapter. Students will revisit this survey after they complete the chapter to see if their perceptions have changed. Leveled Chapter Tests Assessment Options The assessment masters in the Chapter 10 Resource Masters offer a wide range of assessment tools for formative (monitoring) assessment and summative (final) assessment. • Form 1 contains multiple-choice questions and is intended for use with below grade level students. • Forms 2A and 2B contain multiple-choice questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations. • Forms 2C and 2D contain free-response questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations. • Form 3 is a free-response test for use with above grade level students. All of the above mentioned tests include a free-response Bonus question. Quizzes Four free-response quizzes offer assessment at appropriate intervals in the chapter. Mid-Chapter Test This 1-page test provides an option to assess the first half of the chapter. It parallels the timing of the Mid-Chapter Quiz in the Student Edition and includes both multiple-choice and free-response questions. Vocabulary Test This test is suitable for all students. It includes a list of vocabulary words and questions to assess students’ knowledge of those words. This can also be used in conjunction with one of the leveled chapter tests. Extended-Response Test Performance assessment tasks are suitable for all students. Sample answers are included for evaluation. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Standardized Test Practice These three pages are cumulative in nature. It includes two parts: multiple-choice questions with bubble-in answer format and short-answer free-response questions. Answers • The answers for the Anticipation Guide and Lesson Resources are provided as reduced pages. • Full-size answer keys are provided for the assessment masters. Chapter 10 v Glencoe Precalculus NAME DATE 10 PERIOD This is an alphabetical list of key vocabulary terms you will learn in Chapter 10. As you study this chapter, complete each term’s definition or description. Remember to add the page number where you found the term. Add these pages to your Precalculus Study Notebook to review vocabulary at the end of the chapter. Vocabulary Term Found on Page Definition/Description/Example anchor step arithmetic means arithmetic sequence binomial coefficients Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Binomial Theorem common difference common ratio converge diverge Euler’s Formula (Oi-lers) (continued on the next page) Chapter 10 1 Glencoe Precalculus Chapter Resources Student-Built Glossary NAME DATE 10 PERIOD Student-Built Glossary Vocabulary Term Found on Page Definition/Description/Example geometric means geometric sequence inductive hypothesis nth partial sum Pascal’s triangle Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. principle of mathematical induction sequence series sigma notation trigonometric series Chapter 10 2 Glencoe Precalculus NAME 10 DATE PERIOD Anticipation Guide Step 1 Chapter Resources Sequences and Series Before you begin Chapter 10 • Read each statement. • Decide whether you Agree (A) or Disagree (D) with the statement. • Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure). STEP 1 A, D, or NS STEP 2 A or D Statement 1. A sequence can be finite or infinite. 2. The Greek letter Σ is used to indicate a sum. 3. If a sequence has a limit, it is said to diverge. 4. In an arithmetic sequence, the differences between consecutive terms are constant. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 5. If the second differences in a sequence are constant, a cubic function best models the sequence. 6. To find a common ratio in a geometric sequence, multiply any term by the previous term. 7. Some infinite geometric series have a sum. 8. When proving a conjecture using mathematical induction, showing that something works for the first case is called the anchor step. 9. In Pascal’s triangle, the number in row 0 is 0. 10. You can use Euler’s Formula to express a complex number in exponential form. Step 2 After you complete Chapter 10 • Reread each statement and complete the last column by entering an A or a D. • Did any of your opinions about the statements change from the first column? • For those statements that you mark with a D, use a piece of paper to write an example of why you disagree. Chapter 10 3 Glencoe Precalculus NOMBRE 10 FECHA PERÍODO Ejercicios preparatorios Sucesiones y series Paso 1 Antes de que comiences el Capítulo 10 • Lee cada enunciado. • Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado. • Escribe A o D en la primera columna O si no estás seguro(a), escribe NS (no estoy seguro(a)). PASO 1 A, D o NS PASO 2 AoD Enunciado 1. Una sucesión puede ser finita o infinita. 2. La letra griega ∑ se usa para indicar suma. 3. Si una sucesión tiene límite, se dice que diverge. 4. En una sucesión aritmética, la diferencia entre términos consecutivos es constante. 5. Si en una sucesión las segundas diferencias son constantes, entonces se puede representar mejor la sucesión usando una función cúbica. 7. Es posible hacer la suma de algunas series geométricas infinitas. 8. Cuando se prueba una conjetura por inducción matemática, demostrar que algo sí se cumple para un primer caso se llama paso base. 9. En el triángulo de Pascal, el número en la fila 0 es 0. 10. Se puede usar la fórmula de Euler para expresar un número complejo en forma exponencial. Paso 2 Después de que termines el Capítulo 10 • Relee cada enunciado y escribe A o D en la última columna. • Compara la última columna con la primera. ¿Cambiaste de opinión sobre alguno de los enunciados? • En los casos en que hayas estado en desacuerdo con el enunciado, escribe en una hoja aparte un ejemplo de por qué no estás de acuerdo. Capítulo 10 4 Precálculo de Glencoe Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 6. Para calcular la razón común de una sucesión geométrica, multiplica cualquiera de los términos por el término previo. NAME DATE 10-1 PERIOD Study Guide and Intervention Sequences, Series, and Sigma Notation Sequences A sequence is a function with a domain that is the set of natural numbers. The terms of a sequence are the range elements of the function. The nth term is written a n. A term in a recursive sequence depends on the previous term. In an explicit sequence, any nth term can be calculated from the formula. A sequence that approaches a specific value is said to be convergent. Otherwise, it is divergent. Lesson 10-1 Example 1 Find the next four terms of the sequence -7, -3, 4, 14, 27, … . Find the difference between terms to determine a pattern. a 2 − a 1 = −3 − (−7) = 4 a 3 − a 2 = 4 − (−3) = 7 The differences are increasing by 3. a 4 − a 3 = 14 − 4 = 10 a 5 − a 4 = 27 − 14 = 13 The next four terms are 43, 62, 84, and 109. Example 2 1 + 2. Find the sixth term of the sequence a n = − 2n Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. The sequence is explicit. Substitute 6 for n. 1 1 + 2 = 2− or 2.083 a6 = − 2(6) 12 −a Example 3 n−1 Find the third term of the sequence a 1 = 9, a n = − . 3 The sequence is recursive. The first term is given. You need to find the second term before you can find the third term. −a −9 a 2 = −1 = − or −3 Substitute 2 for n. 3 3 −a 2 −(−3) a 3 = − = − or 1 3 3 Substitute 3 for n. The third term is 1. Exercises 1. Find the next four terms of the sequence 125, 25, 5, 1, … . Find the specified term of each sequence. 2. tenth term; a n = 3n - 7 1 3. third term; a 1 = 3, a n = − n3 − 1 4. eighth term; a n = − 5. fourth term; a 1 = 7, a n = 2a n - 1 + 5 2 Chapter 10 2a n - 1 5 Glencoe Precalculus NAME DATE 10-1 PERIOD Study Guide and Intervention (continued) Sequences, Series, and Sigma Notation Series and Sigma Notation A series is the sum of all the terms of a sequence. The nth partial sum is the sum of the first n terms. A partial sum can be symbolized as Sn. Therefore, S5 is the sum of the first five terms of a sequence. A series may be written using sigma notation, denoted by the Greek letter sigma ∑. A formula is written to the right of sigma. The first number to be substituted for the variable in this formula is given below sigma and the last number to be substituted for the variable is above sigma. The results of each substitution are then added. k ∑ an = a1 + a2 + a3 + … + ak n=1 The starting value of the variable is not always 1. Example 1 Find the seventh partial sum of -22, -10, 1, 11, … . Find the pattern of the sequence to find the fifth, sixth, and seventh terms. Notice that a 2 − a 1 = 12, a 3 − a 2 = 11, a 4 − a 3 = 10. Continuing the pattern: a 5 = 11 + 9 = 20 a 6 = 20 + 8 = 28 a 7 = 28 + 7 = 35 The seventh partial sum is S 7 = −22 + (−10) + 1 + 11 + 20 + 28 + 35 or 63. 4 Example 2 n 21 1 a1 = − or − 2 23 a3 = − or 2 22 or 1 a2 = − 24 a4 = − or 4 4 4 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 2 Find the sum of the series ∑ − . 4 n=1 Find a 1, a 2, a 3, and a 4. 4 4 4 15 2n 1 =− +1+2+4=− ∑ − n=1 4 2 2 Exercises 1. Find the sixth partial sum of a n = 4n − 1. −2a 5 n−1 2. Find the fourth partial sum of a n = − , a 1 = −1. Find each sum. 7 6 2 3. ∑ n - 2 n=3 Chapter 10 (2) 1 4. ∑ 3 − n=1 6 n−2 Glencoe Precalculus NAME 10-1 DATE PERIOD Practice Sequences, Series, and Sigma Notation Find the specified term of each sequence. n2 − n 1. ninth term, an = − 4n − 18 n 2. fourth term, a 1 = 10, a n = (−1) a n − 1 + 5 Determine whether each sequence is convergent or divergent. n (−1) 2n − 1 4. a n = − Lesson 10-1 3. 20, 18, 14, 8, … Find the indicated sum for each sequence. 5. seventh partial sum of 13, 22, 31, 40, … 6. S 4 of a n = 2(3.5) n Find each sum. 5 7. ∑ (n 2 − 2 n) 3 8. ∑ (2n - 3) n=3 n=0 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Write a recursive formula and an explicit formula for each sequence. 9. -4, -1, 4, 11, … 1 3 9 27 10. − , − , − , −, … 2 2 2 2 Write each series in sigma notation. The lower bound is given. 11. 3 + 6 + 9 + 12 + 15; n = 1 12. 24 + 19 + 14 + … + (–1); n = 0 13. SAVINGS Kathryn started saving quarters in a jar. She began by putting two quarters in the jar the first day and then she increased the number of quarters she put in the jar by one additional quarter each successive day. a. Use sigma notation to represent the total number of quarters Kathryn had after 10 days. b. Find the sum represented in part a. Chapter 10 7 Glencoe Precalculus NAME 10-1 DATE PERIOD Word Problem Practice Sequences, Series, and Sigma Notation 1. PUMP A vacuum pump removes 15% of the air from an inflated air mattress on each stroke of its piston. The air mattress contains 20 liters of air before the pump starts. 4. ART The number of cubes in an art sculpture, from top to bottom, is given by the sequence 6, 12, 18, 24, … . a. Write an explicit and a recursive formula for the sequence. a. Write the first three terms of the sequence representing the amount of air, in liters, that remains in the mattress after each stroke of the piston. b. There are 8 rows in the sculpture. Write two series for the number of cubes in the sculpture. One with sigma notation and one without. b. Write an explicit and a recursive formula for the sequence. c. How many cubes are in the sculpture? c. Does the sequence converge or diverge? 2. TEMPERATURE The air temperature in degrees Fahrenheit on a certain hiking trail is given by the formula a n = 85 - 3.5(n - 1), where n is the elevation above sea level, in thousands of feet. Write a recursive formula that can be used to find the temperature. a. Write the sequence representing the triangular numbers. Give the first 10 terms. 3. MONEY A salesman’s commission plan entitles him to ten dollars more than the cube of the sale number for his first five sales. How would you represent the salesman’s total commission after his first five sales using sigma notation? How much would he earn in all for the sales? Chapter 10 b. What is the fifth partial sum of the sequence? c. Find an explicit formula to represent the sequence. 8 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 5. GEOMETRY Triangular numbers can be represented by triangles. The first four triangular numbers are 1, 3, 6, and 10. NAME 10-1 DATE PERIOD Enrichment Solving Equations Using Sequences You can use sequences to solve many equations. For example, consider x 2 + x - 1 = 0. You can proceed as follows. x2 + x - 1 = 0 Original equation 2 x +x=1 Add 1 to each side. x(x + 1) = 1 Factor. 1 x=− Divide each side by (x + 1). x+1 1 Next, define the sequence a 1 = 0 and a n = − . 1 + an - 1 The limit of the sequence is a solution to the original equation. 1 1. Let a 1 = 0 and a n = − . 1 + an - 1 a. Write the first five terms of the sequence. Do not simplify. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. b. Write decimals for the first five terms of the sequence. c. Use a calculator to compute a 6, a 7, a 8, and a 9. Compare a 9 with the positive solution of x 2 + x - 1 = 0 found by using the quadratic formula. 2. Use the method described above to find a root of 3x 2 - 2x - 3 = 0. 3. Open a spreadsheet. Type 0 in cell B1. Type = 3/(3*B1-2) in cell B2. Press enter and the cell displays -1.5. Drag the contents of this cell to B50. When do the terms stop changing? Compare this method to the method in Exercise 2. Chapter 10 9 Glencoe Precalculus NAME 10-2 DATE PERIOD Study Guide and Intervention Arithmetic Sequences and Series Arithmetic Sequences Arithmetic sequences are formed when the same number is added to each term to make the next term. The constant amount added to each term is the common difference. The common difference is found by subtracting any term from the term that follows it. To calculate the nth term of an arithmetic sequence, use the formula a n = a 1 + (n − 1) d, where a 1 is the first term of the sequence and d is the common difference. Arithmetic means are terms between two nonconsecutive terms in an arithmetic sequence. Example 1 Find the 38th term of the arithmetic sequence -7, -5, -3, … . First find the common difference. a 2 − a 1 = −5 − (−7) or 2 a 3 − a 2 = −3 − (−5) or 2 Use the explicit formula a n = a 1 + (n - 1) d to find a 38. Use n = 38, a 1 = -7, and d = 2. a 38 = -7 + (38 - 1)2 = 67 Exercises 1. Find the 100th term of the arithmetic sequence 1.6, 2.3, 3, … . 2. Find the 28th term of the arithmetic sequence -1, -3, -5, … . 3. Find the first term of the arithmetic sequence for which a 15 = 30 and d = 1.4. 4. Find d in the arithmetic sequence for which a 1 = 6 and a 40 = 142.5. 5. Write an arithmetic sequence that has three arithmetic means between 17 and 39. 6. Write an arithmetic sequence that has seven arithmetic means between -2 and 16. Chapter 10 10 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Example 2 Write an arithmetic sequence that has three arithmetic means between 3.2 and 4.4. ? ? ? The sequence will have the form 3.2, , , , 4.4. Find d. a n = a 1 + (n -1) d Formula for nth term of arithmetic sequence 4.4 = 3.2 + (5 - 1) d Substitute. 4.4 = 3.2 + 4d Simplify. d = 0.3 Determine the arithmetic means recursively. a 2 = 3.2 + 0.3 = 3.5, a 3 = 3.5 + 0.3 = 3.8, a 4 = 3.8 + 0.3 = 4.1 The sequence is 3.2, 3.5, 3.8, 4.1, 4.4. NAME DATE 10-2 PERIOD Study Guide and Intervention (continued) Arithmetic Sequences and Series Arithmetic Series An arithmetic series is the sum of the terms of an arithmetic sequence. You can use a formula to find the sum of a finite arithmetic series or the partial sum of an infinite arithmetic series. n If you know the first and last terms, a 1 and a n, use the formula S n = − (a 1 + a n). 2 If you know the first term and the common difference, a 1 and d, use n Sn = − [2a 1 + (n − 1) d]. 2 Example 1 Find the sum of the first 50 terms in the series 11 + 14 + 17 + … + 158. n Because the first and last terms are known, use S n = − (a 1 + a n). 2 Substitute 50 for n, 11 for a 1, and 158 for a 50. Lesson 10-2 50 S 50 = − (11 + 158) 2 = 4225 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Example 2 Find the 23rd partial sum of the arithmetic series 173 + 162 + 151 + … . The 23rd term in not known. The first term is known and the common difference n can be found by subtracting 162 - 173 = -11. Use S n = − [2a 1 + (n − 1)d]. 2 S 23 23 =− [2(173) + (23 − 1)(−11)] 2 = 1196 Exercises 1. Find the 82nd partial sum of the arithmetic series -1 + (-4) + (-7) + … . 2. Find the sum of the first 25 terms in the series 7 + 10 + 13 + … + 79. 3. Find the 53rd partial sum of the arithmetic series 12 + 20 + 28 + … . 4. Find the sum of the first 42 terms in the series 1.5 + 2 + 2.5 + … + 22. 15 5. Find ∑ (3n + 1). n=3 42 6. Find ∑ 2n. n=1 7. Find a quadratic model for the sequence 8, 16, 26, 38, 52, 68, … . Chapter 10 11 Glencoe Precalculus NAME 10-2 DATE PERIOD Practice Arithmetic Sequences and Series Determine the common difference, and find the next four terms of each arithmetic sequence. 1. -1.1, 0.6, 2.3, … 2. 16, 13, 10, … Find both an explicit formula and a recursive formula for the nth term of each arithmetic sequence. 3. 9, 13, 17, … 4. 75, 70, 65, … Find the specified value for the arithmetic sequence with the given characteristics. 5. If a 1 = -27 and d = 3, find a 24. 6. If a n = 27, a 1 = -12, and d = 3, find n. 7. If a 23 = 32 and a 1 = -12, find d. 8. If a 6 = 5 and d = -3, find a 1. Find the indicated arithmetic means for each set of nonconsecutive terms. 10. 2 means; -7 and 2.75 Find the indicated sum of each arithmetic series. 11. S 13 of -5 + 1 + 7 + … + 67 12. 62nd partial sum of -23 + (-21.5) + (-20) + … 21 13. Find the sum ∑ (-6n + 4). n= 5 14. Find a quadratic model for the sequence 6, 11, 18, 27, 38, 51, … . 15. DESIGN Wakefield Auditorium has 26 rows. The first row has 22 seats. The number of seats in each row increases by 4 as you move to the back of the auditorium. a. How many seats are in the last row? b. What is the seating capacity of this auditorium? 16. WORK The first-year salary of an employee is $34,500. Each year thereafter, her annual salary increases by $750. a. What will her salary be during her tenth year of work? b. What will her total earnings be for 25 years of work? Chapter 10 12 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 9. 3 means; 35 and 45 NAME 10-2 DATE PERIOD Word Problem Practice Arithmetic Sequences and Series 1. CONSTRUCTION A retaining wall is being built out of bricks. The bottom row of the wall has 150 bricks. Each row contains 5 fewer bricks than the row below it. How many bricks should be ordered if the wall is to be 20 rows tall? 5. READING Resolving to read more each week, Drew starts a new reading program and promises to read 10 minutes the first week, 20 minutes the second, 30 minutes the third, and so on. a. How much time will Drew spend reading during the 12th week? Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 3. COMMISSION A company will give Roberto $100 for the first sale he makes. Each sale after that, they will give him an extra $40.50 per sale. So, he will make $140.50 for the second sale, $181 for the third sale, and so on. How many sales will he have to make in a month to earn at least $2000? b. After one year, how much time did Drew spend reading? 6. CARS Professional drivers can accelerate very quickly. The times and distances for a racing car are listed in the table below. Times (s) 1 2 3 4 5 Distance (m) 15 60 135 240 375 a. Calculate the first and second differences of the sequence. b. What type of model best describes this sequence? c. Find the model for this sequence. 4. SALARY An employee agreed to a salary plan where her annual salary increases by the same amount each year. If she earned $50,100 for the third year and $57,300 for the seventh year, how much was her pay for the first year? Chapter 10 13 Glencoe Precalculus Lesson 10-2 2. GARDENING Alison bought 10 peonies to start a flowerbed. In the fall, she splits the plants, which results in her getting 4 more peonies each year. If she continues to do this every year, how many peonies will Alison have in 10 years? NAME 10-2 DATE PERIOD Enrichment Writing Figurative Numbers as Finite Arithmetic Series Triangular numbers are numbers that can be represented by a triangle using that same number of dots. The first three triangular numbers are 1, 3, and 6. 1. Write an expression using sigma to represent the nth triangular number. Explain your reasoning. (Hint: Consider the number of extra dots needed to make the next triangular number from the previous triangular number.) Likewise, square numbers are numbers that can be represented by a square using that same number of dots. 2. Write an expression using sigma to represent the nth square number. Explain your reasoning. Pentagonal numbers can be represented by a regular pentagon using that same number of dots. 4. Write an expression using sigma to represent the nth pentagonal number. Explain your reasoning. 5. The first five hexagonal numbers are 1, 6, 15, 28, and 45. Write an expression using sigma to represent the nth hexagonal number. Explain your reasoning. 6. Study the pattern in the sigma notations for your answers. Use it to predict the first 5 heptagonal (seven-sided) numbers. Explain your reasoning. Chapter 10 14 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 3. The first five pentagonal numbers are 1, 5, 12, 22, and 35. Use dots to show why 12 is the third pentagonal number. NAME 10-3 DATE PERIOD Study Guide and Intervention Geometric Sequences and Series Geometric Sequences A geometric sequence is a sequence in which each term after the first, a1, is the product of the preceding term and the common ratio, r. Therefore, to find the common ratio, divide any term by the previous term. Any nth term can be calculated with the formula an = a1r n -1. The terms between two nonconsecutive terms of a geometric sequence are called geometric means. Write a sequence that has two geometric means between 6 and 162. Example 2 ? ? The sequence will resemble 6, _____, _____, 162. This means that n = 4, a1 = 6, and a4 = 162. Find r. an = a1r n - 1 Formula for nth term of a geometric sequence 3 162 = 6r Substitute. 3 27 = r Divide each side by 6. 3=r Take the cube root of each side. Determine the geometric means recursively. a2 = 6(3) or 18, a3 = 18(3) or 54 The sequence is 6, 18, 54, 162. Exercises 1. Determine the common ratio and find the next three terms of the geometric sequence x, 2x, 4x, … . 2. Find the seventh term of the geometric sequence 157, -47.1, 14.13, … . 3. Find the 17th term of the geometric sequence 128, 64, 32, … . 4. Find the first term of the geometric sequence for which a6 = 0.1 and r = 0.2. 5. Find r of the geometric sequence for which a1 = 15 and a10 = 7680. 6. Write a geometric sequence that has three arithmetic means between 7 and 567. Chapter 10 15 Glencoe Precalculus Lesson 10-3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Find the seventh term of the geometric sequence 8, -24, 72, … . Example 1 First, find the common ratio. a2 ÷ a1 = -24 ÷ 8 or -3 a3 ÷ a2 = 72 ÷ (-24) or -3 Use the explicit formula an = a1(r) n - 1 to find a7. Use n = 7, a1 = 8, and r = -3. a7 = 8 (-3)7 - 1 = 5832 NAME DATE 10-3 Study Guide and Intervention PERIOD (continued) Geometric Sequences and Series Geometric Series A geometric series is the sum of the terms of a geometric sequence. You can use a formula to find the sum of a finite geometric series or the partial sum of an infinite geometric series. a -ar 1 n If you know the first and last terms, a1 and an, use the formula Sn = − . 1-r 1 - rn If you know the first term and the number of terms, a1 and n, use Sn = a1 − . 1-r a1 ) ( An infinite geometric series converges if |r| < 1 and its sum is given by S = − . 1-r Example 1 Find the sum of the first 12 terms of the geometric series 6 + 7.5 + 9.375 + … . The common ratio is 7.5 ÷ 6 or 1.25. Because the first term and number of (1-r) 1 - 1.25 = 6( − 1 - 1.25 ) 1 - rn terms is known, use Sn = a1 − . Substitute 12 for n, 6 for a1, and 1.25 for r. S12 12 ≈ 325.246 Example 2 If possible, find the sum of the geometric series 40 + 8 + 1.6 + … . a 1 S=− 1-r 40 =− or 50 1 - 0.2 Exercises 1. Find the sum of the first seven terms of -1 + (-4) + (-16) + … . 9 2. Find the sum of a geometric series if a1 = 8, and an = 0.394, and r = − . 11 11 3. Find ∑ 5 (1.06)n - 1. n=1 4. Find the sum of the first 16 terms in a geometric series where a1 = 1, and an = -2an - 1. If possible, find the sum of each infinite geometric series. ∞ () 3 5. ∑ 13 − 8 n=1 Chapter 10 n-1 ∞ 6. ∑ 3n - 1 n=1 16 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. The common ratio is 8 ÷ 40 or 0.2. Because |0.2| < 1, the series has a sum. NAME 10-3 DATE PERIOD Practice Geometric Sequences and Series Determine the common ratio and find the next three terms of each geometric sequence. 9 2. -4, -3, - − ,… 1. -1, 2, -4, … 4 Write an explicit formula and a recursive formula for the nth term of each geometric sequence. 3. 2, 10, 50, … 4. 12, –18, 27, … 5. a5 for 20, 0.2, 0.002, … 1 1 6. a3 for a6 = − ,r=− 7. a1 for a4 = 28, r = 2 8. a9 for √ 3 , -3, 3 √ 3, … 32 2 Lesson 10-3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Find the specified term for each geometric sequence or sequence with the given characteristics. Find the indicated geometric means for each pair of nonconsecutive terms. 9. 2 and 0.25; 2 means 10. -32 and -2; 3 means Find each sum. 3 9 27 11. first eight terms of − +− +− +… 4 20 100 12. a1 = -3, an = 786,432, r = -4 11 13. ∑ -2 (1.5)n-1 6 14. ∑ 3 (0.2)n-1 n=3 n=2 If possible, find the sum of each infinite geometric series. ∞ 15. 10 + 5 + 2.5 + … (3) 1 16. ∑ 6 − n=2 n-1 17. POPULATION A city of 100,000 people is growing at a rate of 5.2% per year. Assuming this growth rate remains constant, estimate the population of the city five years from now. Chapter 10 17 Glencoe Precalculus NAME 10-3 DATE PERIOD Word Problem Practice Geometric Sequences and Series 1. ACCOUNTING Each year, the value of a car depreciates by 18%. If you bought a $22,000 car in 2009, what will be its value in 2015? 5. SCIENCE Bismuth-210 has a half-life of 5 days. This means that half of the original amount of the substance decays every five days. Suppose a scientist has 250 milligrams of Bismuth-210. a. Complete the table to show the amount of Bismuth-210 every five days. 2. BACTERIA A colony of bacteria grows at a rate of 10% per day. If there were 100,000 bacteria on a surface initially, about how many bacteria would there be after 30 days? Half-Life 0 1 2 3 4 Day 0 5 10 15 20 Amount (mg) 250 b. The amounts of Bismuth-210 can be written as a sequence with the half-life number as the domain. Write an explicit and recursive formula for finding the nth term of the geometric sequence. Source: U.S. Census Bureau c. How much Bismuth-210 will the scientist have after 50 days? Round to the nearest hundredth. d. Graph the function that represents the sequence. 4. SALARY An employee agreed to a salary plan where his annual salary increases by 4.5% each year. He earned $50,081.41 for his tenth year of work. 140 y 120 a. What was his pay for his first year of work? 100 80 60 40 b. To the nearest dollar, how much did he earn for his first 10 years of work? 20 0 Chapter 10 18 2 4 6 8 10x Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 3. POPULATION From 1990 to 2000, Florida’s population grew by about 23.5%. The population in the 2000 census was 15,982,378. If this rate of growth continues, what will be the approximate population in 2030? NAME 10-3 DATE PERIOD Enrichment Installment Loans Many installment loans, including home mortgages, credit card purchases, and some car loans, compute interest only on the outstanding balance. Part of each equal payment goes for interest and the remainder reduces the amount owed. As the outstanding balance decreases, the amount of interest paid each term decreases. Let A represent the amount borrowed, p the amount of each payment, I the interest rate, n the number of payments, and bk the balance after k payments. The first balance b1 equals the amount borrowed A plus the interest A(I) minus one payment p. The second balance b2 equals b1 plus the interest b1(I) minus another payment p and so on. Fill in the blanks. b1 = A + AI - p = A(1 + I) - p b2 = b1 + b1I - p = b1(1 + I) - p or A(1 + I)2 - p(1 + I) - p b3 = b2 + b2I - p = b2(1 + I) - p or and r = . 1 - (1 + I)n -I The formula for the sum of a geometric series gives Sn = = − (1 + I)n - 1 I or − for the expression in brackets. Since the last balance bn equals 0, [ (1 + I)n - 1 I ] 0 = A(1 + I)n - p − . Solving for p gives p = , AI . which simplifies to a formula for determining the monthly payment, p = − -n 1 - (1 + I) If payments are made monthly, then I is the monthly interest rate and n is the total number of monthly payments. Find the amount of the monthly payment for each loan. 1. $6000 at 15% per year for four years 2. $75,000 at 13% per year for thirty years 3. $1200 at 18% per year for nine months 4. $11,500 at 9.5% per year for five years Chapter 10 19 Glencoe Precalculus Lesson 10-3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Continue this pattern to the nth balance. bn = A(1 + I)n - p[(1 + I)n - 1 + (1 + I)n - 2 + . . . + (1 + I) + 1] The expression in brackets is a geometric series with a = NAME 10-3 DATE PERIOD TI-Nspire Activity Finding Terms and Sums You can generate terms of a sequence in the LISTS & SPREADSHEET application. Example 1 Find the 15th term of the sequence an = 4 (2)n - 1. Add a LISTS & SPREADSHEET page. Move the cursor to the gray cell above A1 and select MENU > DATA > GENERATE SEQUENCE. Type the formula 4(2)n - 1 next to u(n) =, enter 4 as the initial term, and enter 15 as the maximum number of terms. Be sure to use parentheses to indicate the exponent. Tab down to OK and press ·. Column A is now populated with the first 15 terms of the sequence. The 15th term is 65,536. You can find the sum of the first n terms in either the LISTS & SPREADSHEET application or in the CALCULATOR application. Example 2 Find the sum of the first 15 terms of the sequence in Example 1. Method 1: These are the terms in column A. Move to cell B1 and type the formula = SUM(A1 : A15) and press ·. The sum is 131,068. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Method 2: Insert a CALCULATOR page. Press the catalog key and choose ∑ (. As shown at the bottom of the Catalog page, the format is (expression, variable, low, high). Type 4(2)n - 1, n, 1, 15) and press ·. Exercises 1. Find the 7th term of the sequence an = 3.5(3)n - 1. 2. Find the 10th term of the sequence an = 5(2)n - 1. 3. Find the 15th term of the sequence an = 500 (0.5)n - 1. 4. Find the sum of the first 7 terms of the sequence in Exercise 1. 5. Find the sum of the first 10 terms of the sequence in Exercise 2. 6. Find the sum of the first 15 terms of the sequence in Exercise 3. (4) 1 7. Find the sum of the first 30 terms of the sequence an = 24 − Chapter 10 20 n-1 . Glencoe Precalculus NAME DATE 10-4 PERIOD Study Guide and Intervention Mathematical Induction Mathematical Induction A method of proof called mathematical induction can be used to prove certain conjectures and formulas. A conjecture can be proven true if you can show that something works for the first case, assume that it works for any particular case, and then show it works for the next case. Example Use mathematical induction to prove that the sum of the first n positive even integers is n(n + 1). Here P n is defined as 2 + 4 + 6 + . . . + 2n = n(n + 1). Step 1: First, verify that P n is true for the first possible case, n = 1. Since the first positive even integer is 2 and 1(1 + 1) = 2, the formula is true for n = 1. Step 2: Then assume that Pn is true for n = k. Pk ⇒ 2 + 4 + 6 + . . . + 2k = k(k + 1). Replace n with k. Next, prove that Pn is also true for n = k + 1. Pk + 1 ⇒ 2 + 4 + 6 + . . . + 2k + 2(k + 1) = k(k + 1) + 2(k + 1) Add 2(k + 1) to both sides. We can simplify the right side by adding k(k + 1) + 2(k + 1). (k + 1) is a common factor. If k + 1 is substituted into the original formula (n(n + 1)), the same result is obtained. (k + 1)[(k + 1) + 1] or (k + 1)(k + 2) Thus, if the formula is true for n = k, it is also true for n = k + 1. Since Pn is true for n = 1, it is also true for n = 2, n = 3, and so on. That is, the formula for the sum of the first n positive even integers is true for all positive integers n. Exercise Lesson 10-4 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Pk + 1 ⇒ 2 + 4 + 6 + . . . + 2k + 2(k + 1) = (k + 1)(k + 2) 1. Use mathematical induction to prove that 1 + 5 + 9 + 13 + . . . + (4n - 3) = n(2n - 1) is true for all positive integers n. Chapter 10 21 Glencoe Precalculus NAME DATE 10-4 Study Guide and Intervention PERIOD (continued) Mathematical Induction Extended Mathematical Induction The extended principle of mathematical induction is used when a statement is not true for n = 1. The first step is to prove that P n is true for the first possible case. Example Prove that n! > 5 n for integer values of n ≥ 12. Step 1: Let P n be the statement that n! > 5 n for integer values n ≥ 12. The first possible case is n = 12. Verify that P n is true for n = 12. 12! = 479,001,600 and 5 12 = 244,140,625 and 479,001,600 > 244,140,625. Step 2: Assume P n is true for n = k, so assume k! > 5 k for some positive integer k > 12. Show that (k + 1)! > 5 k + 1 is true. k! > 5 k Inductive hypothesis (k + 1) · k! > (k + 1) · 5 k Multiply each side by k + 1. (k + 1)! > (k + 1) · 5 k Definition of factorial For k > 12, we know that k + 1 > 5. The Multiplication Property of Inequality states we can multiply each side of an inequality by a positive value and maintain the inequality. Therefore, we can multiply each side of k + 1 > 5 by 5 k to obtain (k + 1) · 5 k > 5 · 5 k. (k + 1)! > (k + 1) · 5 k > 5 · 5 k Combined inequality (k + 1)! > 5 · 5 k Transitive Property of Inequality (k + 1)! > 5 k + 1 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Therefore, (k + 1)! > 5 Simplify using a property of exponents. k+1 is true. Because P n is true for n = 12 and for n = k + 1, it is true for all integers n ≥ 12. Exercise 1. Prove that 4 n > 4n for n ≥ 2. Chapter 10 22 Glencoe Precalculus NAME DATE 10-4 PERIOD Practice Mathematical Induction Use mathematical induction to prove that each conjecture is valid for all positive integers n. n(n + 1) 3 n 1 2 1. − +− +− + ... + − =− 3 3 3 6 2. 5 n + 3 is divisible by 4. Lesson 10-4 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 3 Chapter 10 23 Glencoe Precalculus NAME 10-4 DATE PERIOD Word Problem Practice Mathematical Induction 1. GEOMETRY Diagonals are segments that join nonconsecutive vertices. The number of diagonals in a convex n(n − 3) polygon with n sides is equal to − . 2 2. GRAVEL The gravel at a stone center is sold in 5-pound increments. Customers can load their trucks by using either 10-pound or 25-pound buckets. Prove that all gravel sales greater than 25 pounds can be loaded using just the 10- and 25-pound buckets. a. What is the least possible value of n? b. Explain why for every additional vertex added to the polygon, the number of diagonals increases by n - 1. Chapter 10 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. c. Use the extended principle of mathematical induction to prove the statement above. 24 Glencoe Precalculus NAME 10-4 DATE PERIOD Enrichment Conjectures and Mathematical Induction Frequently, the pattern in a set of numbers is not immediately evident. Once you make a conjecture about a pattern, you can use mathematical induction to prove your conjecture. f (x) 1. a. Graph f(x) = x 2 and g(x) = 2 x on the axes shown at the right. b. Write a conjecture that compares n 2 and 2 n, where n is a positive integer. x 0 2. Refer to the diagrams at the right. a. How many dots would there be in the fourth diagram S4 in the sequence? 4 4 4 b. Describe a method that you can use to determine the number of dots in the fifth diagram S 5 based on the number of dots in the fourth diagram, S 4. Verify your answer by constructing the fifth diagram. Lesson 10-4 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. c. Use mathematical induction to prove your response from part b. c. Find a formula that can be used to compute the number of dots in the nth diagram of this sequence. Use mathematical induction to prove your formula is correct. Chapter 10 25 Glencoe Precalculus NAME 10-5 DATE PERIOD Study Guide and Intervention The Binomial Theorem Pascal’s Triangle In Pascal’s triangle, the first and last numbers in each row is 1 and the number in row 0 is 1. Other numbers are the sum of the two numbers above them. The first five rows of Pascal’s triangle are shown below. Row 0 1 Row 1 1 Row 2 1 Row 3 Row 4 1 1 1 2 3 1 3 4 6 1 4 1 The numbers in Pascal’s triangles are the binomial coefficients when (a + b)n is expanded. You can use these numbers to expand binomials without multiplying repeatedly. The first term is an, the last term is bn, and the powers of a decrease by 1 as the powers of b increase by 1 from left to right. Example Use Pascal’s triangle to expand (x + 2y) 5. First, write the series for (a + b)5 without coefficients. Then replace a with x and b with 2y. a5b0 + a4b1 + a3b2 + a2b3 + a1b4 + a0b5 Series for (a + b)5 x5 (2y)0 + x4 (2y)1 + x3 (2y)2 + x2 (2y)3 + x1 (2y)4 + x0 (2y)5 Substitution 5 4 3 2 2 3 4 5 x + x (2y) + x (4y ) + x (8y ) + x (16y ) + 32y Simplify. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. The numbers in the fifth row of Pascal’s triangle are the coefficients. Following the pattern above, these numbers will be 1, 5, 10, 10, 5, and 1. 1x5 + 5x4 (2y) + 10x3 (4y2) + 10x2 (8y3) + 5x (16y4) + 1 · 32y5 Add coefficients. 5 4 3 2 2 3 4 5 x + 10x y + 40x y + 80x y + 80xy + 32y Simplify. Exercises Use Pascal’s triangle to expand each binomial. 1. (x + 4)3 2. (3x + y)4 3. (7 + g)4 4. (m - n)6 5. (2a – 2b)5 6. (c + d)7 Chapter 10 26 Glencoe Precalculus NAME DATE 10-5 Study Guide and Intervention PERIOD (continued) The Binomial Theorem The Binomial Theorem The binomial coefficient of the an - r br term in the expansion of (a + b) is given by nCr. You can find nCr by using a n n! . calculator or by finding − (n - r)! r! The Binomial Theorem states that for any positive integer n, the expansion of (a + b)n is C anb0 + nC1 an - 1b1 + nC2 an - 2b2 + … + nCr an - rbr + … + nCn a0bn. n 0 Example 1 Find the coefficient of the fourth term in the expansion of (5a + 2b)6. For (5a + 2b)6 to have the form (a + b)n, let a = 5a and b = 2b. Since r increases from 0 to n, r is one less than the number of the term. Evaluate 6C3. 6! 6! · 5 · 4 · 3! C3 = − =− = 6− or 20 6 (6 - 3)!3! 3!3! 3!3! Example 2 Use the Binomial Theorem to expand (3x + 7)4. Let a = 3x and b = 7. (3x + 7)4 = 4C0(3x)4(7)0 + 4C1(3x)3(7)1 + 4C2(3x)2(7)2 + 4C3(3x)1(7)3 + 4C4(3x)0(7)4 = 1 · 81x4 · 1 + 4 · 27x3 · 7 + 6 · 9x2 · 49 + 4 · 3x · 343 + 1 · 1 · 2401 = 81x4 + 756x3 + 2646x2 + 4116x + 2401 Exercises Find the coefficient of the indicated term in each expansion. 1. (x + 5)6, fourth term 2. (3a + 4b)8, a3b5 term Use the Binomial Theorem to expand each binomial. 3. (x + 3)5 4. (4x + 2y)3 5. (x - 2y)4 6. (2x - 3y)4 Chapter 10 27 Lesson 10-5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. The binomial coefficient of the fourth term in (a + b)6 is 20. Substitute for a and b in an - rbr. 20(5a)6 – 3(2b)3 = 20(5a)3(2b)3 = 20(125a3)(8b) = 20,000a3b The coefficient is 20,000. Glencoe Precalculus NAME 10-5 DATE PERIOD Practice The Binomial Theorem Use Pascal’s triangle to expand each binomial. 1. (r + 3)5 2. (3a + b)4 Find the coefficient of the indicated term in each expansion. 3. (2n - 3m)4, 4th term 4. (4a + 2b)8, 5th term 5. (3p + q)9, q5p4 term 6. (a - 2 √ 3 ) , 3rd term 6 Use the Binomial Theorem to expand each binomial. 7. (x - 5)4 5 10. (2p - 3q)6 11. Represent the expansion of (3x + 8y)15 using sigma notation. 12. SPORTS A varsity volleyball team needs nine members. Of these nine members, at least five must be seniors. How many of the possible groups of juniors and seniors have at least five seniors? Chapter 10 28 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 9. (a - √ 2) 8. (3x + 2y)4 NAME DATE 10-5 PERIOD Word Problem Practice The Binomial Theorem 1. GOLF A golfer can drive a ball to the fairway about 70% of the time. What is the probability of hitting the fairway on exactly 14 of the 18 holes? 4. SPORTS On average, a basketball player misses 3 free throws out of every 8 attempts. If the player attempts 5 free throws, what is the probability that he misses no more than two times? 2. FAMILY Suppose a mother and father have 6 children. Assume that having a girl or boy are equally likely outcomes. 5. FOOD The probability that an apple does not meet the quality-control standards for continuing down an assembly line to become filling for a pie is 4.5%. A batch of 75 apples is received. a. Complete the table to show the probability that they have each number of girls. Number of Girls a. What is the probability that 5 or fewer of the apples will be rejected? Probability 0 1 b. What is the probability that at least 72 of the apples go into a pie? 2 4 5 c. What is the probability that all 75 apples go into a pie? 6 b. What is the probability that at least four of the children are girls? 6. WORK The probability that a substitute teacher has to work on a Friday during any given week in a certain school district is 32%. What is the probability that the substitute teacher will work on three of the four Fridays in the upcoming month? 3. PROMOTION A juice company is holding a promotion where one in every five bottles of juice has a coupon for a free bottle of juice. If a customer buys three bottles, what is the probability that all three bottles have a free juice coupon? Chapter 10 Lesson 10-5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 3 29 Glencoe Precalculus NAME 10-5 DATE PERIOD Enrichment Patterns in Pascal’s Triangle You have learned that the coefficients in the expansion of (x + y)n yield a number pyramid called Pascal’s triangle. This activity explores some of the interesting properties of this famous number pyramid. 1. Pick a row of Pascal’s triangle. a. What is the sum of all the numbers in all the rows above the row that you picked? b. What is the sum of all the numbers in the row that you picked? c. How are your answers for parts a and b related? d. Repeat parts a through c for at least three more rows of Pascal’s triangle. What generalization seems to be true? Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. e. See if you can prove your generalization. 2. Pick any row of Pascal’s triangle that comes after the first. a. Starting at the left end of the row, find the sum of the odd numbered terms. b. In the same row, find the sum of the even numbered terms. c. How do the sums in parts a and b compare? d. Repeat parts a through c for at least three other rows of Pascal’s triangle. What generalization seems to be true? Chapter 10 30 Glencoe Precalculus NAME DATE 10-6 PERIOD Study Guide and Intervention Functions as Infinite Series Power Series 1 The rational function f(x) = − 1-x ∞ can be expressed as the infinite series ∑ x or 1 + x + x2 + … + x n n n=0 for|x| < 1. ∞ A power series in x is an infinite series of the form ∑ anxn = a0 + a1x + n=0 Example Lesson 10-6 10-1 a2x2 + a3x3 + …, where x and a can take on any values n = 0, 1, 2, … . ∞ Use ∑ xn to find a power series representation of n=0 1 . Indicate the interval on which the series converges. g(x) = − 1 - 3x Use a graphing calculator to graph g(x) together with the sixth partial sum of its power series. ∞ f(x) = ∑ xn for |x| < 1 and g(x) is a transformation of f(x). To find the n=0 transformation, write g(x) = f(u) and solve for u. Here, g(x) = f(3x). ∞ Replace x with 3x in f(x) to get f(3x) = ∑ (3x)n for |3x| < 1. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. n=0 1 1 The series converges for |3x| < 1, which can be written as - − <x<− . 3 3 Find the sixth partial sum. 5 ∑ (3x)n or 1 + 3x + (3x)2 + (3x)3 + (3x)4 + (3x)5 n=0 1 and S6(x) are shown. The graphs of g(x) = − [-0.5, 0.5] scl: 0.1 by [-1, 6] scl: 1 1 - 3x Exercise ∞ 1 1. Use ∑ x n to find a power series representation of g(x) = − . 2-x n=0 Indicate the interval on which the series converges. Use a graphing calculator to graph g(x) together with the sixth partial sum of its power series. Chapter 10 31 Glencoe Precalculus NAME DATE 10-6 PERIOD Study Guide and Intervention (continued) Functions as Infinite Series Transcendental Functions as Power Series The value of ex can be approximated by using the exponential series. The trigonometric series can be used to approximate values of the trigonometric functions. Euler’s Formula can be used to write the exponential form of a complex number that is the natural logarithm of a negative number. ∞ Exponential Series xn x2 x3 x4 x5 e =∑ − =1+x+− +− +− +− +… x n=0 2! n! ∞ 3! 4! 5! n 2n (-1) x x2 x4 x6 x8 +− -− +− -… cos x = ∑ − = 1 - − Trigonometric Series n=0 ∞ (2n)! 2! 4! 6! 8! n 2n + 1 (-1) x x3 x5 x7 x9 sin x = ∑ − = x - − +− -− +− -… n=0 (2n + 1)! 3! 5! 7! 9! iθ a2 + b2 and Exponential Form of a a + bi = re , where r = √ b -1 b Complex Number θ = tan-1 − a for a > 0 and θ = tan − a + π for a < 0 Example Use the fifth partial sum of the trigonometric series π for sine to approximate the value of sin − . Round to three decimal 6 places. x3 x5 x7 x9 sin x = x - − +− -− +− -… π sin − 6 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Let x 3! 5! 7! 9! π = − or about 0.524. 6 (0.524)3 (0.524)5 (0.524)7 (0.524)9 ≈ 0.524 - − + − - − + − 3! 5! 7! 9! ≈ 0.500 Exercises 1. Write 4 - 4i in exponential form. 2. Use the fifth partial sum of the exponential series to approximate the value of e2.7. Round to three decimal places. 3. Write 2 √ 3 + 2i in exponential form. 4. Use the fifth partial sum of the trigonometric series for cosine to π approximate the value of cos − . Round to three decimal places. 8 Chapter 10 32 Glencoe Precalculus NAME DATE 10-6 PERIOD Practice Functions as Infinite Series ∞ 2 1. Use ∑ xn to find a power series representation of g(x) = − . 3-x n=0 Indicate the interval on which the series converges. Use a graphing calculator to graph g(x) together with the sixth partial sum of its power series. 2. e 0.5 Lesson 10-6 10-1 Use the fifth partial sum of the exponential series to approximate each value. Round to three decimal places. 3. e 1.2 Use the fifth partial sum of the trigonometric series for cosine or sine to approximate each value. Round to three decimal places. 5π 4. sin − − 5. cos 3π 6 4 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Write each complex number in exponential form. π π 6. 13 cos − + i sin − ( 3 3 ) 8. 1 - √ 3i 7. 5 + 5i Find the value of each natural logarithm in the complex number system. 9. ln (-4) 10. ln (-5.7) 11. ln (-1000) 12. SAVINGS Derika deposited $500 in a savings account with a 4.5% interest rate compounded continuously. (Hint: The formula for continuously compounded interest is A = Pert.) a. Approximate Derika’s savings account balance after 12 years using the first four terms of the exponential series. b. How long will it take for Derika’s deposit to double, provided she does not deposit any additional funds into her account? Chapter 10 33 Glencoe Precalculus NAME 10-6 DATE PERIOD Word Problem Practice Functions as Infinite Series 1. INVESTMENT Jill deposits $2000 into an account that compounds continuously at 3.0%. 4. ENDANGERED SPECIES The bald eagle was placed on the endangered species list in 1967 and removed in 2007. The number of breeding pairs in the lower 48 states is documented below. a. Write a power series to approximate Jill’s account balance, assuming she does not deposit any more money. 1963 1974 1984 1990 1994 2000 2006 Pairs 487 791 1757 3035 4449 6471 9789 a. Determine an exponential regression equation for this data. Use the year number for x. b. Use the first five terms of the series to find the amount of money in the account after 5 years. 2. MECHANICS The function π f(x) = 10 cos − x models the distance in 12 centimeters a weight on a spring is from its initial position after x seconds, without regard for friction. Use the fifth partial sum of the trigonometric series for cosine to find the distance after 2 seconds. ( Year b. Approximate the number of breeding pairs in 2012. ) 3. STOCKS An analyst notices that the early growth of a stock price in hundreds of dollars per share can be modeled by 1 P(x) = − x , where x is time in months. 4-− 4 a. Write a power series approximation for the price of this stock. Where does it converge? d. Use your power series to the sixth term to approximate the number of breeding pairs of bald eagles in 2012. Is this a good approximation? Why or why not? b. Compare the power series approximation after 3 and 6 terms to the original equation for x = 14. Chapter 10 34 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. c. By using the logarithmic change of base formula, you can write the exponential equation with a base e. f(x) = ab x = e x ln b + ln a. Write a power series to approximate your regression equation from part a. NAME 10-6 DATE PERIOD Enrichment Alternating Series The series below is called an alternating series. 1-1+1-1+… The reason is that the signs of the terms alternate. An interesting question is whether the series converges. In the exercises, you will have an opportunity to explore this series and others like it. Lesson 10-6 10-1 1. Consider 1 - 1 + 1 - 1 + … . a. Write an argument that suggests that the sum is 1. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. b. Write an argument that suggests that the sum is 0. c. Write an argument that suggests that there is no sum. (Hint: Consider the sequence of partial sums.) If the series formed by taking the absolute values of the terms of a given series is convergent, then the given series is said to be absolutely convergent. It can be shown that any absolutely convergent series is convergent. 2. Create an alternating series, other than a geometric series with negative common ratio, that has a sum. Justify your answer. Chapter 10 35 Glencoe Precalculus NAME DATE 10-6 PERIOD Graphing Calculator Activity Approximating Sine Using Polynomial Functions In this activity you will examine polynomial functions that can be used to approximate sin x. x3 x5 Graph f(x) = sin x and g(x) = x - − +− on the 3! same screen. Press — ( 4 ) 3 ÷ 3 + ENTER MATH 5 ÷ 5 ( 4 MATH 5! ) SIN Y= ) ENTER GRAPH [–9.42, 9.42] scl: 1 by [–2, 2] scl: 1 . Step 1 Use TRACE to help you write an inequality describing the x-values for which the graphs seem very close together. Press TRACE and use and to move along the graph of y = sin x. Press or x3 x5 to move the cursor to the graph of y = x - − +− . Press and 3! 5! to move along the graph. In absolute value, what are the greatest and least differences between the values of f(x) and g(x) for the values of x described by the inequality that you wrote? x3 x5 x7 Step 2 Repeat Step 1 using h(x) = x - − +− -− instead of g(x). 3! 5! 7! x3 x5 x7 x9 Step 3 Repeat Step 1 using k(x) = x - − +− -− +− . 3! 5! 7! 9! [–9.42, 9.42] scl: 1 by [–2, 2] scl: 1 Exercises 1. Are the intervals for which you get good approximations for sin x larger or smaller for polynomials that have more terms? 2. What term should be added to k(x) to obtain a polynomial with six terms that gives good approximations to sin x? Chapter 10 36 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. [–9.42, 9.42] scl: 1 by [–2, 2] scl: 1 NAME DATE 10 PERIOD Chapter 10 Quiz 1 SCORE (Lessons 10-1 and 10-2) 1. Find the sixth term of the sequence an = n2 − n. 1. 2. Does the sequence 8, 6, 4, 2, … converge or diverge? 2. 6 3. Find the sum of the series ∑ 2n − 4. 3. n=1 4. Find the common difference of the sequence 19.82, 28.39, 36.96, … . 4. 5. If a1 = 1000 and d = –4, find a52. 5. 61 6. MULTIPLE CHOICE What is the sum of ∑ (0.2n + 2.6)? n=1 B 528 C 536.8 D 1073.6 6. 7. Find S22 of the series 0 + 1.3 + 2.6 + … . 7. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAME DATE 10 PERIOD Chapter 10 Quiz 2 SCORE (Lesson 10-3) 1. Find the fourth term of the geometric sequence an = −2an − 1, a1 = −1. 1. 2. Find the sum of the first six terms of 1 + 1.5 + 2.25 + … . 2. 3. Write an explicit formula for finding the nth term of the sequence 2, 10, 50, 250, … . 3. 4. Write a sequence that has two geometric means between 3 and 12. 4. ∞ (3) 1 5. Find the sum of the series ∑ − n=1 n−1 . 5. 6. MULTIPLE CHOICE What is the common ratio of the series 1 1 1 1 − +− +− +− +…? 10 20 1 A − 20 Chapter 10 40 80 1 B − 10 1 C − 3 1 D − 2 37 6. Glencoe Precalculus Assessment A 451.4 NAME 10 DATE PERIOD Chapter 10 Quiz 3 SCORE (Lessons 10-4 and 10-5) 1. MULTIPLE CHOICE What is the correct order for the Principle of Mathematical Induction? A Anchor Step, Inductive Hypothesis, Inductive Step B Inductive Hypothesis, Anchor Step, Inductive Step C Inductive Hypothesis, Inductive Step, Anchor Step D Anchor Step, Inductive Step, Inductive Hypothesis 1. 2. Suppose that in a proof of the summation formula 5 + 11 + 17 + … + (6n −1) = n(3n − 2) by mathematical induction, it has already been shown valid for n = 1. Also, the assumption of validity for some n = k is complete. Write the next step in the induction step of this proof. 2. 3. Use Pascal’s triangle to expand (h + k)4. 3. 4. Use the Binomial Theorem to find the coefficient for the fourth term of the expansion of (3z − d)8. 4. 5. PRIZES The probability that Kiyoto wins a prize in a cereal box is 0.3. What is the probability that he wins exactly 2 prizes when buying 3 boxes? 5. 10 DATE PERIOD Chapter 10 Quiz 4 (Lesson 10-6) 1 1. Write the power series that is equivalent to f (x) = − . 9−x Indicate the interval on which the series converges. 1. 2. Use the fifth partial sum of the exponential series to approximate e4.1 to the nearest hundredth. 2. π 3. Write the first four terms of the power series of sin − and use 3 it to approximate the value. 3. 4. Write 4i in exponential form. 4. 5. MULTIPLE CHOICE What is the complex value of ln (–8)? A iπ - ln 8 Chapter 10 B i ln 8 C iπ + ln 8 38 D π + i ln(8) 5. Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAME NAME DATE 10 PERIOD Chapter 10 Mid-Chapter Test SCORE (Lessons 10-1 through 10-3) Part I Write the letter for the correct answer in the blank at the right of each question. 1. What are the next two terms of the sequence 1, 5, 11, 19, …? A 21, 25 B 29, 41 C 27, 35 D 25, 35 1. 2. Which shows the series 0 + 2 + 6 + 12 written in sigma notation? 4 4 H ∑ (n 2 - 1) F ∑ (2n - 2) n=1 n=1 4 4 G ∑ (n - 1)(n - 2) J ∑ n(n -1) n=1 2. n=1 A -27.5 B 15 C 12.5 D 27.5 3. H 15, 13 J 16, 11 4. C 867 D 892.5 5. 4. Find two arithmetic means between the terms 18 and 9. F 15, 12 G 16, 14 34 3n ? 5. What is the sum of the series ∑ − n=1 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A 433.5 B 446.25 4 Part II Write the answer in the blank at the right of each question. 6. PARADOX To travel 10 miles, you must first cover 5 miles. To travel the next 5 miles, you must first cover the next 2.5 miles. Express this travel paradox where the first two terms are 5 and 2.5 in sigma notation. 6. 1 3 5 7. Determine if the sequence − , −, −, … is arithmetic, geometric, 5 7 9 or neither. Then find the next term. 7. 8. Write the explicit and recursive formula for the sequence -4, -1, 2, … . 8. 2n + 1 n −n 9. Find the fifth term of an = − . 3 9. 10. Determine whether 80,000, 8000, 800, … converges or diverges. 10. Chapter 10 39 Glencoe Precalculus Assessment 3. If a12 = -15 and d = -2.5, find a1. NAME 10 DATE PERIOD Chapter 10 Vocabulary Test anchor step arithmetic means arithmetic sequence arithmetic series binomial coefficients Binomial Theorem common difference common ratio converge SCORE power series recursive sequence second difference sequence series sigma notation term trigonometric series geometric means geometric sequence geometric series inductive hypothesis inductive step infinite sequence infinite series nth partial sum diverge Euler’s Formula explicit sequence exponential series Fibonacci sequence finite sequence finite series first difference Choose the correct term from the list above to complete each sentence. 1. 2. A(n) __________ gives an as a function of n and does not require any previous terms. 2. 3. A(n) __________ is the sum of the terms of a sequence in which the difference between successive terms is constant. 3. 4. The __________ are terms between two known, nonconsecutive terms of a geometric sequence. 4. 5. If a sequence has a limit such that the terms approach a unique number, then it is said to __________. 5. 6. The coefficients of the expansion of (a + b)n are called __________. 6. 7. A(n) __________ is the sum of all of the terms of a finite or an infinite sequence. 7. 8. The __________ is a well-known recursive sequence that describes many patterns of numbers found in nature. 8. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 1. A sequence in which the ratio between successive terms is constant is called a(n) __________. Define each term in your own words. 9. Euler’s Formula 10. recursive sequence Chapter 10 40 Glencoe Precalculus NAME DATE 10 PERIOD Chapter 10 Test, Form 1 SCORE Write the letter for the correct answer in the blank at the right of each problem. 1. Express the series 5 + 9 + 13 + … + 101 using sigma notation. ∞ 25 A ∑ (4n + 1) B ∑ (4n + 1) n=1 25 C ∑ (4n − 1) n=1 24 D ∑ (4n + 1) n=1 1. n=1 2. Find the next two terms of the sequence 8, 2, -4, … . F -8, -12 G -10, -16 H 10, 16 J -6, -8 2. D 2816 3. 3. Find the fifth term in the sequence 11, -44, 176, … . A -2816 B -704 C 704 4. The next term in the Fibonacci sequence 1, 1, 2, 3, 5, … is ____. G 7 H 8 J 15 4. 5. Find the 15th term in the arithmetic sequence 14, 10.5, 7, … . A -63 B -35 C 63 D 66.5 5. 6. In an arithmetic sequence, what is d if a1 is 13 and a71 = 223? G 6 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. F -3 H 3 J -2 6. 7. Find the sum of the first 20 terms in the arithmetic series 14 + 3 - 8 + … . A -1810 B -195 C 195 D 1810 7. 8. SALARY An employee agreed to a salary plan where his annual salary increases by the same amount each year. If he earned $49,310 for the fourth year and $65,310 for the ninth year, how much was his pay for the first year? F $18,200 G $39,710 H $42,910 J $46,110 8. 4 9. Write ∑ 3 k-1 in expanded form and then find the sum. k =1 A 1 + 3 + 9 + 27; 40 C 3 + 9 + 27 + 81; 120 1 1 1 40 B 1+− +− +− ;− D 0 + 2 + 8 + 26; 36 3 9 27 27 9. 10. Which are the two geometric means between 2 and -1024? F -8, 8 G -6, -14 H -16, 128 J 255.5, 511 C 50 D does not exist 11. 10. 11. Find the sum of 22 + 11 + 5.5 + … . A 40 B 44 12. APPRECIATION Each year, the value of an antique increases by 6%. If the antique was worth $1600 in 2009, what will its value be in 2015? F $1174.25 Chapter 10 G $1677.22 H $2141.16 41 J $2269.63 12. Glencoe Precalculus Assessment F 6 NAME DATE 10 Chapter 10 Test, Form 1 PERIOD (continued) 13. Suppose in a proof of the summation formula 1 + 5 + 9 + ... + 4n - 3 = n(2n - 1) by mathematical induction, you show the formula valid for n = 1 and assume that it is valid for n = k. What is the next equation in the induction step of this proof? A 1 + 5 + 9 + ... + 4k - 3 + 4(k + 1) - 3 = k(2k - 1) + 4(k + 1) - 3 B 1 + 5 + 9 + ... + 4k - 3 = k(2k - 1) + 4(k + 1) - 3 C 1 + 5 + 9 + ... + 4k - 3 = k(2k - 1) D 1 + 5 + 9 + ... + 4k - 3 + 4(k + 1) - 3 = k(2k - 1) + (k + 1)[2(k + 1) - 1] 13. 14. What is the third term in the expansion (x + 4y)4? F 64y3 G 48x2y2 H 96x2y2 14. J 256xy3 15. The expression 32x5 + 80x4 + 80x3 + 40x2 + 10x + 1 is the expansion of which binomial? A (2x + 1)5 B (x + 2)5 C (2x + 2)5 D (2x - 1)5 15. 16. PRIZES The probability of choosing a yogurt with a winning lid is 0.25. What is the approximate probability that exactly 2 of the 4 yogurts Shirley bought have winning lids? F 2.3% G 6.3% H 21.0% J 28.1% 16. B iπ - 3.0445 C iπ + 3.0445 D -3.0445 17. 17. What is ln (-21)? 18. What is 1 - i in exponential form? 7π π i− i− 2e 4 G √ 2e 4 F √ H ei 7π − 4 π − 18. J ei 4 19. INVESTMENT Ms. Tirado puts $1800 into an account that compounds continuously at 2.0%. Which series can be used to approximate the account balance, assuming she does not deposit any more money? ∞ n (0.02x) 1800n! A ∑ − n=0 ∞ n=0 ∞ (0.02x)n B 1800∑ − n! n=0 n x C 1800∑ − ∞ 0.02n! (0.02x)n n! D 1800∑ − n=0 19. 1 ? 20. Which is the power series representation of f(x) = − ∞ (2x)n F ∑ − n! n=0 ∞ H ∑ (2x)n n=0 ∞ (-1)n (2x - 5)2n + 1 G ∑ −− (2n + 1)! n=0 6 - 2x ∞ J ∑ (2x - 5)n 20. n=0 Bonus If a1, a2, a3, ... , an is an arithmetic sequence, where an ≠ 0, 1 1 1 1 then − a1 , − a2 , − a3 , ... , − an is a harmonic sequence. Find one harmonic mean between 2 and 3. Chapter 10 42 B: Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. A 3.0445 NAME DATE 10 PERIOD Chapter 10 Test, Form 2A SCORE Write the letter for the correct answer in the blank at the right of each problem. 1. Express the series 0.7 + 0.007 + 0.00007 + … using sigma notation. ∞ ∞ A ∑ 0.7(10) n−1 ∞ B ∑ 7(10) n=1 1 − 2n ∞ C ∑ 7(10) 1 − 2n D ∑ 0.7(10)−n n=0 n=1 1. n=1 2. Find the next two terms of the sequence 10, -11, -32, … . F -53, -74 G -43, -54 H -42, -53 J -22, -12 3 y3, -3y5, 3 √ 3 y7, … . 3. Find the next term in the sequence √ y7 B 9 √3 C 9y9 D - √ 3 y2 A -9y9 2. 3. 4. The ninth term in the Fibonacci sequence 1, 1, 2, 3, 5, … is ___. G 34 H 55 J 144 4. 5. Find the 27th term in the arithmetic sequence -8, 1, 10, … . A 174 B 226 C 235 D 242 5. 6. In an arithmetic sequence, what is d if a1 is 14 and a24 = 50.8? Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. F 1.6 G 2.1 H 2.6 J 3.6 6. 7. Find the sum of the first 36 terms in the arithmetic series -0.2 + 0.3 + 0.8 + … . A 318.6 B 332.2 C 307.8 D 315 7. 8. SALARY An employee’s salary increases by the same amount each year. If he earned $77,900 for the seventh year and $97,500 for the fifteenth year, how much was his pay for the second year? F $61,100 4 G $63,200 (3) 2 9. Write ∑ 5 − k=2 2 k B 2 2 3 (3) 1 (3) 2 2 C 5 − 15,700 52 52 52 + (− + (− ;− (− 3 ) 3 ) 3 ) 81 2 J $65,650 8. in expanded form and then find the sum. 28 ( 3 ) + (−23 ) + (−23 ) ; − 9 2 A 5 − H $63,900 4 2 D 5 − (3) 2 (3) 3 2 +5 − 2 +5 − 2 3 190 +5 − ;− (3) 27 2 4 380 +5 − ;− (3) 81 9. 10. Which are the two geometric means between 175 and 1.4? F 0.2, 0.04 G 1.4, 0.0112 H 35, 7 J 131.25, 65.625 10. 1( C − 9 - 9 √ 3) D does not exist 11. 27 + √ 9 + √ 3+…. 11. Find the sum of √ 1( A − 9 + 9 √ 3) 2 Chapter 10 B 9 + 9 √ 3 2 43 Glencoe Precalculus Assessment F 13 NAME DATE 10 Chapter 10 Test, Form 2A PERIOD (continued) 12. APPRECIATION Each year, the value of a trading card increases by 4.8%. If the card was worth $155 in 2009, what will its value be in 2021? F $247.71 G $259.60 H $272.06 12. J $285.12 13. Suppose in a proof of 7 + 9 + 11 + ... + 2n + 5 = n(n + 6) by mathematical induction, you show the formula valid for n = 1. Assume that it is valid for n = k. What is the next equation in this proof? A 7 + 9 + 11 + . . . + 2k + 5 + 2(k + 1) + 5 = k(k + 6) + (k + 1)(k + 1 + 6) B 7 + 9 + 11 + . . . + 2(k + 1) + 5 = k(k + 6) C 7 + 9 + 11 + . . . + 2k + 5 = k(k + 6) D 7 + 9 + 11 + . . . + 2k + 5 + 2(k + 1) + 5 = k(k + 6) + 2(k + 1) + 5 13. 14. What is the fifth term in the expansion (3x − 2y)6? F 240xy4 G -32y5 H -576xy5 J 2160x2y4 14. 15. The expression 243c5 + 810c4d + 1080c3d2 + 720c2d3 + 240cd4 + 32d5 is the expansion of which binomial? A (3c + d)5 B (c + 2d)5 C (2c + 3d)5 D (3c + 2d)5 15. 16. SPORTS The probability that Kelly makes a free throw is 0.85. What is the approximate probability that she makes at least 8 of her next 10 attempts? G 68% H 72% J 82% 16. B iπ + 4.6250 C iπ − 4.6250 D -4.6250 17. 17. What is ln (-102)? A 4.6250 3 - 15i in exponential form? 18. What is 15 √ F 30e 11π i− G 30e 6 5π i− H 30e 6 7π i− J 15e 6 11π i− 18. 6 19. INVESTMENT Ms. Bing puts $3500 into an account that compounds continuously at 3.2%. Which can be used to approximate the balance? ∞ ∞ (0.032x)n 3500n! n=0 ∞ n=0 ∞ (0.032x)n n! B 3500∑ − n=0 n x C 3500∑ − A ∑ − 0.032n! (0.032x)n n! D 3500∑ − n=0 19. 3 20. On which interval does the power series f(x) = − converge? 6 − 5x F 0<x<2 Bonus Chapter 10 6 G 0<x<− 5 5 H − <x<2 3 Find the sum of the coefficients of (x + 2)6. 44 J 3<x<6 20. B: Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. F 61% NAME DATE 10 PERIOD Chapter 10 Test, Form 2B SCORE Write the letter for the correct answer in the blank at the right of each problem. 1. Express the series -27 +9 -3 +1 - … using sigma notation. ∞ 3 ( 3) 1 B ∑ −27 − − A ∑ −3n n=0 n=0 n ∞ ( 3) 1 C ∑ −27 − − n=0 n ∞ ( 3) 1 D ∑ 27 − − n=0 n 1. 2. Find the next two terms of the sequence 14, -5, -24, … . F -33, -42 G -43, -62 H -29, -34 J -15, -6 2. 2 b4, −2b8, 2 √ 2 b12, … . 3. Find the next term in the sequence √ b4 B −4b16 C 4 √2 D −2 √ 2 b4 A −2b16 3. F 21 G 34 J 144 4. D -111 5. H 55 5. Find the 21st term in the arithmetic sequence 9, 3, -3, … . A -129 B -126 C -117 6. In an arithmetic sequence, what is d if a1 is -11 and a51 = 59? Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. F 1.2 G 1.4 H 1.6 J 2 6. 7. Find the sum of the first 20 terms in the arithmetic series -6 -12 -18 - … . A -2520 B -1266 C -1260 D -1140 7. 8. SALARY An employee’s salary increases by the same amount each year. If he earned $61,325 for the sixth year and $87,000 for the nineteenth year, how much was his pay for the second year? F $37,361.67 3 G $39,073.34 ( 2) 1 9. Write ∑ − − k=0 k 4 J $53,425 8. in expanded form and then find the sum. 1 1 3 1 C −− +− −− ;− 7 1 1 1 A −− −− −− ;−− 2 H $51,450 8 4 8 8 1 1 1 5 D 1−−+−−− ;− 2 4 8 8 8 1 1 1 1 B 1 − − − − − −; − 2 4 8 8 2 9. 10. Which are the two geometric means between 192 and 0.375? F 0.375, 0.75 G 24, 3 H 144, 72 J 72, 36 10. 35 C − 7 D − 11. 5 10 20 −− +− −…. 11. Find the sum of − 2 7 A − 9 Chapter 10 14 98 18 B − 7 18 45 2 Glencoe Precalculus Assessment 4. The eighth term in the Fibonacci sequence 1, 1, 2, 3, 5, … is ___. NAME DATE 10 Chapter 10 Test, Form 2B PERIOD (continued) 12. APPRECIATION Each year, the value of a trading card increases by 3.9%. If it was worth $210 in 2010, what will its value be in 2018? F $274.49 G $285.20 H $296.32 12. J $296.70 1 n (5 - 1) by mathematical 13. Suppose in a proof of 1 + 5 + 25 + … + 5n - 1 = − 4 induction, you show the formula valid for n = 1. Assume that it is valid for n = k. What is the next equation in this proof? 1 k 1 k+1 (5 - 1) + − (5 - 1) A 1 + 5 + 25 + … + 5k - 1 + 5k + 1 -1 = − 4 4 1 k (5 - 1) + 5k + 1 - 1 B 1 + 5 + 25 + … + 5k + 5k + 1 = − 4 C 1 + 5 + 25 + … + 5 k-1 D 1 + 5 + 25 + … + 5 k-1 +5 k+1-1 +5 k+1-1 1 k+1 =− (5 - 1) + 5k + 1 - 1 4 1 k =− (5 - 1) + 5k + 1 - 1 4 13. 14. What is the fourth term in the expansion (2x − 5y)4? F -1000xy3 G -125xy3 H 600x2y5 J 625y4 14. 15. The expression 81p4 + 216p3r + 216p2r2 + 96pr3 + 16r4 is the expansion of which binomial? A (3p + 2r)4 B (3p + 4r)4 C (2p + 3r)4 D (4p + 3r)4 15. F 12% G 24% H 28% J 44% 16. B 4.5109 C iπ − 4.5109 D iπ + 4.5109 17. 17. What is ln (−91)? A -4.5109 18. What is 3 − √ 3 i in exponential form? F 9e 11π i− G 9e 6 5π i− 3 H 2 √ 3e 11π i− J 2 √ 3e 6 5π i− 18. 3 19. INVESTMENT Mr. Cook puts $2550 into an account that compounds continuously at 2.9%. Which can be used to approximate the balance? ∞ ∞ (0.029x)n A ∑ − 2550n! n=0 n=0 ∞ ∞ n x B 2550∑ − n=0 (0.029x)n n! C 2550∑ − (0.029x)n n! D 2550∑ − 0.029n! n=0 19. 4 converge? 20. On which interval does the power series f(x) = − 8 − 3x F 4<x<8 3 G − <x<2 4 H 0<x<2 8 J 0<x<− 3 20. 6 Bonus Solve ∑ (3n − 2x) = 7 for x. B: n=0 Chapter 10 46 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 16. SPORTS The probability that Kurt makes a strike is 0.35. What is the approximate probability that he makes at least 2 strikes on the next 3 frames? NAME DATE 10 PERIOD Chapter 10 Test, Form 2C SCORE 1. Express the series 40 - 20 + 10 - 5 using sigma notation. 1. 2. Find the next two terms of the sequence 8, -4, -16… . 2. x2, -5x3, 5 √ 3. Find the next term in the sequence √5 5 x4, … . 3. 4. What is the eleventh term of the Fibonacci sequence 1, 1, 2, 3, 5, … ? 4. Assessment Write the correct answer in the blank at the right of each problem. 5. Find the 15th term of the arithmetic sequence 3 4 2 11 − , 10 − , 9, 7 − ,…. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 5 5 5. 5 6. If a4 is 13 and a34 = 103, find the common difference, d. 6. 7. Find the sum of the first 27 terms of the arithmetic series 35.5 + 34.3 + 33.1 + 31.9 + … . 7. 8. SALARY An employee agreed to a salary plan where her annual salary increases by the same amount each year. If she earned $52,800 for the tenth year and $67,500 for the seventeenth year, how much was her pay for the third year? 8. 7 ( 3) 1 9. Write ∑ 27 − − k=2 k−2 in expanded form. Then find the sum. 10. Write a sequence that has three geometric means between 6 and 54. Chapter 10 47 9. 10. Glencoe Precalculus NAME DATE 10 Chapter 10 Test, Form 2C PERIOD (continued) 11. Find the sum of the first eight terms in the geometric 1 series − + 2 + 20 + … . 5 11. 12. DEPRECIATION Each year, the value of a tractor decreases by 8.5%. If the tractor was worth $55,000 in 2009, what will its value be in 2017? 12. 1 n 13. Suppose that in a proof 1 + 5 + 25 + … + 5n − 1 = − (5 − 1) 4 has already been shown valid for n = 1 by mathematical induction. Also, the assumption of validity for some n = k is complete. Write the next step in the induction step of this proof. 13. 14. Use the Binomial Theorem to expand (6x − y)4. 14. 15. What is the fifth term in the expansion of (3x3 + 2y2)5? 15. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 16. FLOWERS The probability that a flower from a certain pack of seeds blossoms is 0.7. What is probability that at least 3 of 5 randomly chosen seeds from the packet blossom? 16. 17. Find ln (-12.7). Round to four decimal places. 17. + i in exponential form. 18. Write √3 18. 19. INVESTMENT Mr. Harrison puts $4600 into an account that compounds continuously at 3.1%. Write a series that can be used to approximate the account balance, assuming he does not deposit any more money. 19. 3 20. Find the power series representation of f(x) = − . 9 − 7x Bonus Chapter 10 Find the sum of the coefficients of (2x + y)5. 48 20. B: Glencoe Precalculus NAME DATE 10 PERIOD Chapter 10 Test, Form 2D SCORE 1. Express the series 240 - 60 + 15 - 3.75 using sigma notation. 1. 2. Find the next two terms of the sequence 7, -5, -17… . 2. x3, -7x6, 7 √ 3. Find the next term in the sequence √7 7 x9, … . 3. 4. What is twelfth term of the Fibonacci sequence 1, 1, 2, 3, 5, … ? 4. 5. Find the 40th term of the arithmetic sequence 22 9 4 7, − , −, − − ,…. 5. 6. If a1 is 6 and a13 = −42, find the common difference d. 6. 7. Find the sum of the first 30 terms of the arithmetic series 10 + 6.8 + 3.6 + … . 7. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 5 5 5 Assessment Write the correct answer in the blank at the right of each problem. 8. SALARY An employee agreed to a salary plan where his annual salary increases by the same amount each year. If he earned $51,100 for the fifth year and $64,900 for the eleventh year, how much was his pay for the third year? 8. 7 ( 3) 1 9. Write ∑ 81 − − k=2 k−2 in expanded form. Then find the sum. 10. Form a sequence that has three geometric means between 4 and 16. Chapter 10 49 9. 10. Glencoe Precalculus NAME 10 DATE Chapter 10 Test, Form 2D PERIOD (continued) 11. Find the sum of the first eight terms in the geometric series 64 − 32 + 16 − 8… . 11. 12. DEPRECIATION Each year, the value of a tractor decreases by 7.5%. If the tractor was worth $63,000 in 2008, what will its value be in 2019? 12. 13. Suppose that in a proof of the summation formula 7 + 9 + 11 + … + (2n + 5) = n(n + 6) by mathematical induction, it has already been shown valid for n = 1. Also, the assumption of validity for some n = k is complete. Write the next step in the induction step of this proof. 13. 14. Use the Binomial Theorem to expand (2x − 3y)4. 14. 15. What is the fifth term in the expansion of (2x2 + y4)5? 15. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 16. FLOWERS The probability that a flower from a certain pack of seeds blossoms is 0.9. What is probability that at least 5 of 7 randomly chosen seeds from the packet blossom? 16. 17. Find ln (-13.4). Round to four decimal places. 17. 18. Write 1 − i in exponential form. 18. 19. INVESTMENT Ms. Lawrence puts $7300 into an account that compounds continuously at 2.7%. Write a series that can be used to approximate the account balance, assuming she does not deposit any more money. 19. 5 20. Find the power series representation of f(x) = − . 20. 10 − 3x Bonus Chapter 10 Find the sum of the coefficients of (2x − y)4. 50 B: Glencoe Precalculus NAME DATE 10 PERIOD Chapter 10 Test, Form 3 SCORE Write the correct answer in the blank at the right of each problem. 1 1. Express the series -25 −5 -1 − − using sigma notation 5 where the lower bound is n = 2. 1. 2. Find the next two terms of the sequence 414, 138, 46, … . 2. 3. Find the fifth term in the sequence Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 3 x y3, −75 x3y3, 375 √ 3 x5 y3, … . 5 √ 4. What is the 12th partial sum of the Fibonacci sequence 1, 1, 2, 3, 5, … ? 4. 5. Find the 53rd term of the arithmetic sequence 0.5, 0.95, 1.4, 1.85, … . 5. 6. If an = 28, a1 = -22, and d = 5, find n. 6. Assessment 3. 7. Find the sum of the first 30 terms of the arithmetic 3 3 1 − 12 − − 12 − ,…. series -13 − 8 4 7. 8 8. SALARY An employee agreed to a salary plan where his annual salary increases by the same amount each year. If he earned $51,710 for the ninth year and $68,670 for the seventeenth year, how much did he earn in total after 20 years? 8. 8 ⎛ 1 ⎞k − 3 9. Write∑ 128 ⎪− − in expanded form. Then find ⎥ ⎝ 4⎠ k=4 the sum. If the series were infinite, would the terms converge or diverge? 10. Write a sequence that has three geometric means between -4 and -36. Chapter 10 51 9. 10. Glencoe Precalculus NAME DATE 10 Chapter 10 Test, Form 3 PERIOD (continued) 11. Find the sum of the first eleven terms in the geometric 1 1 4 −− +− +…. series − 36 27 11. 81 12. PHYSICS A tennis ball is dropped from a height of 3 of the distance after each fall. 55 feet and bounces − 5 How far does the ball travel upward after it hits the ground the second time? What is the total vertical distance the ball travels before coming to rest? 13. Use the extended principle of mathematical induction to prove that 6n + 24 < n! for n ≥ 5. 12. 13. 14. 15. Find the fifth term in the expansion of (m3 + 3n)8. 15. 16. CARNIVAL The probability that a spinner lands on a heart in a carnival game is 0.25. What is the probability that the spinner lands on a heart at least one time in the next 12 spins? 16. 17. Find ln (-34.72). Round to four decimal places. 17. 18. Write − √ 3 + i in exponential form. 18. 19. INVESTMENT Ms. Goggin puts $1500 into an account that compounds continuously at 1.5%. Write the first 4 terms of a power series approximation of her account balance after 5 years. Find the value. 19. 25 20. Find the power series representation of f(x) = − . 20. 3 −9x Bonus Chapter 10 ⎞5 ⎛3 x − 2⎥ . Find the sum of the coefficients of ⎪− ⎝5 ⎠ 52 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. ⎞4 ⎛1 x - 2y⎥ . 14. Use the Binomial Theorem to expand ⎪− ⎝2 ⎠ B: Glencoe Precalculus NAME DATE 10 PERIOD Chapter 10 Extended-Response Test Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond the requirements of the problem. 1. a. Write a word problem that involves an arithmetic sequence. Write the sequence and solve the problem. Tell what the answer represents. b. Find the common difference and write the nth term of the arithmetic sequence in part a. Assessment c. Find the sum of the first 12 terms of the arithmetic sequence in part a. Explain in your own words why the formula for the sum of the first n terms of an arithmetic series works. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. d. Does the related arithmetic series converge? Why or why not? 2. a. Write a word problem that involves a geometric sequence. Write the sequence and solve the problem. Tell what the answer represents. b. Find the common ratio and write the nth term of the geometric sequence in part a. c. Find the sum of the first 11 terms of the sequence in part a. d. Describe in your own words a test to determine whether a geometric series converges. Does the geometric series in part a converge? 3. a. Explain in your own words how to use mathematical induction to prove that a statement is true for all positive integers. b. Use mathematical induction to prove that the sum of the first n terms of a geometric series is given by the formula a - a rn 1 1 Sn = − , where r ≠ 1. 1-r ( yx √ y √ x ) 6 -− . 4. Find the fourth term in the expansion of − 2 Chapter 10 53 Glencoe Precalculus NAME 10 DATE PERIOD Standardized Test Practice SCORE (Chapters 1–10) Part 1: Multiple Choice Instructions: Fill in the appropriate circle for the best answer. x2 - 25 1. Which is true about the graph of f(x) = − ? x+5 A It has infinite discontinuity. C It has point discontinuity. B It has jump discontinuity. D It is continuous. 2. Which function describes the graph? 1. A B C D 2. F G H J 3. A B C D 4. F G H J 5. A B C D y F f(x) = |x + 1| G f(x) = |x| + 1 H f(x) = |x − 1| x 0 J f(x) = |x| − 1 3. Find the value of c. A 23° B 43.1° C 46.9° D 56.2° 14 9 c 28° F (f - g)(x) = x3 + x + 2 H (f - g)(x) = x3 + 3x + 6 G (f - g)(x) = -x3 - 3x + 2 J (f - g)(x) = -x3 + x + 2 5. Choose the graph of the relation that has an inverse function. y A 0 y B x 0 y C y D x x 0 x 0 6. Find u · v if u = 〈3, 2〉 and v = 〈−5, 1〉. F 13 G -13 H 0 J 7 6. F G H J C 7 D 10 7. A B C D J 6 8. F G H J 7. What is the value of log7 49? 1 A − 2 B 2 8. Find the distance between the points with polar coordinate ⎛ ⎛ π⎞ 2π ⎞ points ⎪5, − ⎥ and ⎪1, − − ⎥. 3 ⎠ ⎝ 3⎠ ⎝ F 12 Chapter 10 G √ 6 H 36 54 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 4. Given f(x) = 2x + 4 and g(x) = x3 + x + 2, what is (f - g)(x)? NAME DATE 10 PERIOD Standardized Test Practice (continued) (Chapters 1–10) 9. What are the dimensions of matrix AB if A is a 2 × 3 matrix and B is a 3 × 7 matrix? A 7×2 B 3×3 C 3×2 D 2×7 9. A B C D 10. F G H J D √ 3 11. A B C D H ⎢ J not possible 12. F G H J C -6e D 6e 13. A B C D 14. F G H J 15. A B C D 16. F G H J 17. A B C D 10. What is the cross product of 〈1, 1, 0〉 and 〈1, 0, 2〉? F 〈2, −2, −1〉 H 〈0, 1, −1〉 G 〈−2, −2, 1〉 J 〈−2, 0, −1〉 A 9 B -3 C 3 ⎡3 5⎤ 12. Find AB if A = [2 1] and B = ⎢ . ⎣0 1⎦ ⎡ 6 11⎤ ⎣ 11 6⎦ Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. F ⎢ G [6 11] ⎡ 6⎤ ⎣ 11⎦ 13. Solve ln 6 − ln x = −1. 1 A −− 6e 1 B − 6e 33π 14. In positive degrees, − is the same as which measure? 4 F 45° G 135° H 225° J 315° 15. What is the third term in the expansion of (4d − 7g)4? A -1792d3g B 4704d2g2 C -5488dg3 D 588dg2 16. Which expression is equivalent to sin (90° - θ)? F sin θ G cos θ H tan θ J sec θ 5 1 17. Find the 17th term of the arithmetic sequence − , 1, − ,…. 3 A 17 − 3 Chapter 10 B 7 C 9 55 3 D 11 Glencoe Precalculus Assessment 11. Solve 3 x - 9 = 9 −x. NAME DATE 10 Standardized Test Practice PERIOD (continued) (Chapters 1–10) Part 2: Short Response Instructions: Write your answers in the space provided. 18. Write an equation of the sine function with amplitude 1, 2π π , phase shift − , and vertical shift 2. period − 3 15 18. 3 6 9 30 12 −− +− −− +…−− 19. Express the series − 3 5 7 9 21 using sigma notation. 19. 20. Solve the system of equations. 3x − y + z = −4 −4x + y − 2z = −1 −x + 3y − z = 10 20. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 21. Show the anchor step for proving 5 + 11 = 17 + … + (6n - 1) = n(3n + 2) by mathematical induction. 21. 22. Write the given parametric equations in rectangular form. y = t2 − 6 22. x = 3t + 5 23. Which type of conic section is represented by the (y − 1)2 36 (x + 2)2 11 equation − + − = 1? 23. 24. Consider the geometric sequence 300, 225, 168.75, … . a. What is the value of r? 24a. b. What is the 14th term in the sequence? 24b. c. What is the sum of all the terms in the sequence? 24c. Chapter 10 56 Glencoe Precalculus Chapter 10 A1 Glencoe Precalculus Before you begin Chapter 10 Sequences and Series Anticipation Guide DATE PERIOD A D A D D A A D A 2. The Greek letter Σ is used to indicate a sum. 3. If a sequence has a limit, it is said to diverge. 4. In an arithmetic sequence, the differences between consecutive terms are constant. 5. If the second differences in a sequence are constant, a cubic function best models the sequence. 6. To find a common ratio in a geometric sequence, multiply any term by the previous term. 7. Some infinite geometric series have a sum. 8. When proving a conjecture using mathematical induction, showing that something works for the first case is called the anchor step. 9. In Pascal’s triangle, the number in row 0 is 0. 10. You can use Euler’s Formula to express a complex number in exponential form. After you complete Chapter 10 A 3 Answers Chapter Resources 12/4/09 2:16:11 PM Glencoe Precalculus • For those statements that you mark with a D, use a piece of paper to write an example of why you disagree. • Did any of your opinions about the statements change from the first column? Chapter 10 STEP 2 A or D 1. A sequence can be finite or infinite. Statement • Reread each statement and complete the last column by entering an A or a D. Step 2 STEP 1 A, D, or NS • Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure). • Decide whether you Agree (A) or Disagree (D) with the statement. • Read each statement. Step 1 10 NAME 0ii_004_PCCRMC10_893811.indd Sec1:3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. DATE Sequences, Series, and Sigma Notation Study Guide and Intervention PERIOD The differences are increasing by 3. −a Substitute 3 for n. Substitute 2 for n. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 5 Chapter 10 255.5 n3 − 1 4. eighth term; a n = − 2 23 2. tenth term; a n = 3n - 7 3 91 Lesson 10-1 3/20/09 6:48:39 PM Glencoe Precalculus 5. fourth term; a 1 = 7, a n = 2a n - 1 + 5 2a n - 1 5 25 125 625 1 1 1 1, − ,− ,− − 1 3. third term; a 1 = 3, a n = − 5 Find the specified term of each sequence. 1. Find the next four terms of the sequence 125, 25, 5, 1, … . Exercises The third term is 1. −9 a 2 = −1 = − or −3 −a 3 3 −a −(−3) a 3 = −2 = − or 1 3 3 n−1 Find the third term of the sequence a 1 = 9, a n = − . 3 The sequence is recursive. The first term is given. You need to find the second term before you can find the third term. Example 3 12 1 1 + 2 = 2− or 2.083 a6 = − 2(6) 2n 1 + 2. Find the sixth term of the sequence a n = − The sequence is explicit. Substitute 6 for n. Example 2 The next four terms are 43, 62, 84, and 109. a 5 − a 4 = 27 − 14 = 13 a 4 − a 3 = 14 − 4 = 10 a 3 − a 2 = 4 − (−3) = 7 a 2 − a 1 = −3 − (−7) = 4 Find the difference between terms to determine a pattern. Example 1 Find the next four terms of the sequence -7, -3, 4, 14, 27, … . A sequence is a function with a domain that is the set of natural numbers. The terms of a sequence are the range elements of the function. The nth term is written a n. A term in a recursive sequence depends on the previous term. In an explicit sequence, any nth term can be calculated from the formula. A sequence that approaches a specific value is said to be convergent. Otherwise, it is divergent. Sequences 10-1 NAME Answers (Anticipation Guide and Lesson 10-1) PERIOD (continued) Sequences, Series, and Sigma Notation Study Guide and Intervention DATE Find the seventh partial sum of -22, -10, 1, 11, … . A2 n 4 2 2 24 a4 = − or 4 4 22 or 1 a2 = − 4 78 Glencoe Precalculus 005_036_PCCRMC10_893811.indd 6 Chapter 10 n=3 3. ∑ n 2 - 2 7 125 Find each sum. 6 n=1 6 (2) 1 4. ∑ 3 − n−1 2. Find the fourth partial sum of a n = − , a 1 = −1. −2a 5 1. Find the sixth partial sum of a n = 4n − 1. Exercises n=1 15 2n 1 =− +1+2+4=− ∑− 4 23 a3 = − or 2 4 21 1 a1 = − or − 4 2 2 Find the sum of the series ∑ − . 4 n=1 Find a 1, a 2, a 3, and a 4. Example 2 4 n−2 16 Glencoe Precalculus 13 11 − = 11.8125 125 87 −− a 7 = 28 + 7 = 35 The seventh partial sum is S 7 = −22 + (−10) + 1 + 11 + 20 + 28 + 35 or 63. a 6 = 20 + 8 = 28 Continuing the pattern: a 5 = 11 + 9 = 20 Find the pattern of the sequence to find the fifth, sixth, and seventh terms. Notice that a 2 − a 1 = 12, a 3 − a 2 = 11, a 4 − a 3 = 10. Example 1 The starting value of the variable is not always 1. n=1 ∑ an = a1 + a2 + a3 + … + ak k A series is the sum of all the terms of a sequence. The nth partial sum is the sum of the first n terms. A partial sum can be symbolized as Sn. Therefore, S5 is the sum of the first five terms of a sequence. A series may be written using sigma notation, denoted by the Greek letter sigma ∑. A formula is written to the right of sigma. The first number to be substituted for the variable in this formula is given below sigma and the last number to be substituted for the variable is above sigma. The results of each substitution are then added. 12/4/09 2:22:37 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 10 Series and Sigma Notation 10-1 NAME -5 417.375 n=0 8. ∑ (2n - 3) 0 n n=0 n=1 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 7 Chapter 10 7 b. Find the sum represented in part a. 65 n=1 ∑ (n + 1) 10 a. Use sigma notation to represent the total number of quarters Kathryn had after 10 days. 13. SAVINGS Kathryn started saving quarters in a jar. She began by putting two quarters in the jar the first day and then she increased the number of quarters she put in the jar by one additional quarter each successive day. ∑ (24 - 5n) 5 Lesson 10-1 12/4/09 2:23:45 PM Glencoe Precalculus 12. 24 + 19 + 14 + … + (–1); n = 0 ∑ 3n 5 11. 3 + 6 + 9 + 12 + 15; n = 1 Write each series in sigma notation. The lower bound is given. 2 2 1 n−1 an = − (3) 2 for n ≥ 2; an = n 2 − 5 2 2 2 1 a1 = − , a n = 3a n − 1; for n ≥ 2 1 3 9 27 10. − , −, −, −, … a 1 = −4, a n = a n − 1 + 2n − 1; 9. -4, -1, 4, 11, … Write a recursive formula and an explicit formula for each sequence. n=3 7. ∑ (n 2 − 2 n) -6 5 Find each sum. n 3 convergent 5. seventh partial sum of 13, 22, 31, 40, … 280 6. S 4 of a n = 2(3.5) n (−1) 2n − 1 4. a n = − Find the indicated sum for each sequence. divergent 3. 20, 18, 14, 8, … PERIOD 2. fourth term, a 1 = 10, a n = (−1) a n − 1 + 5 Determine whether each sequence is convergent or divergent. 4n − 18 n2 − n 1. ninth term, an = − 4 DATE Sequences, Series, and Sigma Notation Practice Find the specified term of each sequence. 10-1 NAME Answers (Lesson 10-1) Chapter 10 A3 3. MONEY A salesman’s commission plan entitles him to ten dollars more than the cube of the sale number for his first five sales. How would you represent the salesman’s total commission after his first five sales using sigma notation? How much would he earn in all for the sales? a 1 = 85, a n = a n - 1 - 3.5 2. TEMPERATURE The air temperature in degrees Fahrenheit on a certain hiking trail is given by the formula a n = 85 - 3.5(n - 1), where n is the elevation above sea level, in thousands of feet. Write a recursive formula that can be used to find the temperature. converge c. Does the sequence converge or diverge? a 1 = 17, a n = 0.85a n - 1 a n = 20(0.85) ; n b. Write an explicit and a recursive formula for the sequence. 17, 14.45, 12.2825, … a. Write the first three terms of the sequence representing the amount of air, in liters, that remains in the mattress after each stroke of the piston. 8 an = − n(n + 1) 2 Glencoe Precalculus Answers Glencoe Precalculus c. Find an explicit formula to represent the sequence. 35 b. What is the fifth partial sum of the sequence? 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, … a. Write the sequence representing the triangular numbers. Give the first 10 terms. 5. GEOMETRY Triangular numbers can be represented by triangles. The first four triangular numbers are 1, 3, 6, and 10. 216 cubes c. How many cubes are in the sculpture? n=1 ∑ 6n 8 6 + 12 + 18 + 24 + 30 + 36 + 42 + 48; b. There are 8 rows in the sculpture. Write two series for the number of cubes in the sculpture. One with sigma notation and one without. a n = 6n; a n = a n - 1 + 6, a 1 = 6 a. Write an explicit and a recursive formula for the sequence. 4. ART The number of cubes in an art sculpture, from top to bottom, is given by the sequence 6, 12, 18, 24, … . PERIOD 3/20/09 6:48:54 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 8 Chapter 10 n=1 ∑ n 3 + 10; $275 5 DATE Sequences, Series, and Sigma Notation Word Problem Practice 1. PUMP A vacuum pump removes 15% of the air from an inflated air mattress on each stroke of its piston. The air mattress contains 20 liters of air before the pump starts. 10-1 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Enrichment DATE 1 + an - 1 1 + an - 1 1+1 1 1+− 1+1 1 1+− 1 1+− Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 9 Chapter 10 9 Sample answer: cell B20; this method is easier because it requires fewer inputs. 3/20/09 6:49:00 PM Glencoe Precalculus PERIOD 3. Open a spreadsheet. Type 0 in cell B1. Type = 3/(3*B1-2) in cell B2. Press enter and the cell displays -1.5. Drag the contents of this cell to B50. When do the terms stop changing? Compare this method to the method in Exercise 2. -0.7208 2. Use the method described above to find a root of 3x 2 - 2x - 3 = 0. 0.625, 0.6154, 0.6190, 0.6176; solution by quadratic formula: 0.6180 c. Use a calculator to compute a 6, a 7, a 8, and a 9. Compare a 9 with the positive solution of x 2 + x - 1 = 0 found by using the quadratic formula. 0, 1, 0.5, 0.6667, 0.6 b. Write decimals for the first five terms of the sequence. 1+0 1+1 1 1 1 1 0, − ,− , − , − a. Write the first five terms of the sequence. Do not simplify. 1 1. Let a 1 = 0 and a n = − . The limit of the sequence is a solution to the original equation. 1 Next, define the sequence a 1 = 0 and a n = − . x+1 You can use sequences to solve many equations. For example, consider x 2 + x - 1 = 0. You can proceed as follows. x2 + x - 1 = 0 Original equation x2 + x = 1 Add 1 to each side. x(x + 1) = 1 Factor. 1 x=− Divide each side by (x + 1). Solving Equations Using Sequences 10-1 NAME Answers (Lesson 10-1) Arithmetic Sequences and Series Study Guide and Intervention DATE A4 Glencoe Precalculus 005_036_PCCRMC10_893811.indd 10 Chapter 10 10 -2, 0.25, 2.5, 4.75, 7, 9.25, 11.5, 13.75, 16 Glencoe Precalculus 6. Write an arithmetic sequence that has seven arithmetic means between -2 and 16. 17, 22.5, 28, 33.5, 39 5. Write an arithmetic sequence that has three arithmetic means between 17 and 39. 4. Find d in the arithmetic sequence for which a 1 = 6 and a 40 = 142.5. 3.5 3. Find the first term of the arithmetic sequence for which a 15 = 30 and d = 1.4. 10.4 2. Find the 28th term of the arithmetic sequence -1, -3, -5, … . -55 1. Find the 100th term of the arithmetic sequence 1.6, 2.3, 3, … . 70.9 Exercises Example 2 Write an arithmetic sequence that has three arithmetic means between 3.2 and 4.4. ? ? ? The sequence will have the form 3.2, , , , 4.4. Find d. a n = a 1 + (n -1) d Formula for nth term of arithmetic sequence 4.4 = 3.2 + (5 - 1) d Substitute. 4.4 = 3.2 + 4d Simplify. d = 0.3 Determine the arithmetic means recursively. a 2 = 3.2 + 0.3 = 3.5, a 3 = 3.5 + 0.3 = 3.8, a 4 = 3.8 + 0.3 = 4.1 The sequence is 3.2, 3.5, 3.8, 4.1, 4.4. Example 1 Find the 38th term of the arithmetic sequence -7, -5, -3, … . First find the common difference. a 2 − a 1 = −5 − (−7) or 2 a 3 − a 2 = −3 − (−5) or 2 Use the explicit formula a n = a 1 + (n - 1) d to find a 38. Use n = 38, a 1 = -7, and d = 2. a 38 = -7 + (38 - 1)2 = 67 Arithmetic sequences are formed when the same number is added to each term to make the next term. The constant amount added to each term is the common difference. The common difference is found by subtracting any term from the term that follows it. To calculate the nth term of an arithmetic sequence, use the formula a n = a 1 + (n − 1) d, where a 1 is the first term of the sequence and d is the common difference. Arithmetic means are terms between two nonconsecutive terms in an arithmetic sequence. PERIOD 3/20/09 6:49:03 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 10 Arithmetic Sequences 10-2 NAME Arithmetic Sequences and Series Study Guide and Intervention DATE (continued) PERIOD 2 2 2 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 11 Chapter 10 11 Lesson 10-2 12/4/09 2:31:24 PM Glencoe Precalculus 7. Find a quadratic model for the sequence 8, 16, 26, 38, 52, 68, … . a n = n 2 + 5n + 2 n=1 6. Find ∑ 2n. 1806 n=3 42 5. Find ∑ (3n + 1). 364 15 4. Find the sum of the first 42 terms in the series 1.5 + 2 + 2.5 + … + 22. 493.5 3. Find the 53rd partial sum of the arithmetic series 12 + 20 + 28 + … . 11,660 2. Find the sum of the first 25 terms in the series 7 + 10 + 13 + … + 79. 1075 1. Find the 82nd partial sum of the arithmetic series -1 + (-4) + (-7) + … . -10,045 Exercises = 1196 2 23 S 23 = − [2(173) + (23 − 1)(−11)] The 23rd term in not known. The first term is known and the common difference n can be found by subtracting 162 - 173 = -11. Use S n = − [2a 1 + (n − 1)d]. Example 2 Find the 23rd partial sum of the arithmetic series 173 + 162 + 151 + … . = 4225 2 50 S 50 = − (11 + 158) Substitute 50 for n, 11 for a 1, and 158 for a 50. n Because the first and last terms are known, use S n = − (a 1 + a n). Example 1 Find the sum of the first 50 terms in the series 11 + 14 + 17 + … + 158. 2 If you know the first term and the common difference, a 1 and d, use n Sn = − [2a 1 + (n − 1) d]. n If you know the first and last terms, a 1 and a n, use the formula S n = − (a 1 + a n). Arithmetic Series An arithmetic series is the sum of the terms of an arithmetic sequence. You can use a formula to find the sum of a finite arithmetic series or the partial sum of an infinite arithmetic series. 10-2 NAME Answers (Lesson 10-2) Chapter 10 Arithmetic Sequences and Series Practice DATE PERIOD -3; 7, 4, 1, -2 2. 16, 13, 10, … a n = 75 + (n - 1)(-5) a 1 = 75; a n = a n - 1 - 5 4. 75, 70, 65, … 20 8. If a 6 = 5 and d = -3, find a 1. 14 6. If a n = 27, a 1 = -12, and d = 3, find n. A5 -3.75, -0.5 10. 2 means; -7 and 2.75 12 Glencoe Precalculus Answers Glencoe Precalculus 12/4/09 2:31:05 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 12 Chapter 10 a. What will her salary be during her tenth year of work? $41,250 16. WORK The first-year salary of an employee is $34,500. Each year thereafter, her annual salary increases by $750. b. What is the seating capacity of this auditorium? 1872 a. How many seats are in the last row? 122 seats 15. DESIGN Wakefield Auditorium has 26 rows. The first row has 22 seats. The number of seats in each row increases by 4 as you move to the back of the auditorium. 14. Find a quadratic model for the sequence 6, 11, 18, 27, 38, 51, … . a n = n 2 + 2n + 3 n= 5 13. Find the sum ∑ (-6n + 4). -1258 21 12. 62nd partial sum of -23 + (-21.5) + (-20) + … 1410.5 11. S 13 of -5 + 1 + 7 + … + 67 403 Find the indicated sum of each arithmetic series. 37.5, 40, 42.5 9. 3 means; 35 and 45 Find the indicated arithmetic means for each set of nonconsecutive terms. 2 7. If a 23 = 32 and a 1 = -12, find d. 42 5. If a 1 = -27 and d = 3, find a 24. Find the specified value for the arithmetic sequence with the given characteristics. a n = 9 + (n - 1)4 a 1 = 9; a n = a n - 1 + 4 3. 9, 13, 17, … Find both an explicit formula and a recursive formula for the nth term of each arithmetic sequence. 1.7; 4, 5.7, 7.4, 9.1 1. -1.1, 0.6, 2.3, … Determine the common difference, and find the next four terms of each arithmetic sequence. 10-2 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 13 Chapter 10 $46,500 4. SALARY An employee agreed to a salary plan where her annual salary increases by the same amount each year. If she earned $50,100 for the third year and $57,300 for the seventh year, how much was her pay for the first year? 13 3. COMMISSION A company will give Roberto $100 for the first sale he makes. Each sale after that, they will give him an extra $40.50 per sale. So, he will make $140.50 for the second sale, $181 for the third sale, and so on. How many sales will he have to make in a month to earn at least $2000? 46 peonies 2. GARDENING Alison bought 10 peonies to start a flowerbed. In the fall, she splits the plants, which results in her getting 4 more peonies each year. If she continues to do this every year, how many peonies will Alison have in 10 years? 2050 bricks 9 sales DATE PERIOD Distance (m) 2 60 3 135 4 240 quadratic a n = 15n 2 Lesson 10-2 12/4/09 4:48:19 PM Glencoe Precalculus c. Find the model for this sequence. 5 375 b. What type of model best describes this sequence? First: 45, 75, 105, 135 Second: 30, 30, 30 a. Calculate the first and second differences of the sequence. 1 15 Times (s) 6. CARS Professional drivers can accelerate very quickly. The times and distances for a racing car are listed in the table below. 3 2 13,780 min or 229 − h b. After one year, how much time did Drew spend reading? 120 min or 2 h a. How much time will Drew spend reading during the 12th week? 5. READING Resolving to read more each week, Drew starts a new reading program and promises to read 10 minutes the first week, 20 minutes the second, 30 minutes the third, and so on. Arithmetic Sequences and Series Word Problem Practice 1. CONSTRUCTION A retaining wall is being built out of bricks. The bottom row of the wall has 150 bricks. Each row contains 5 fewer bricks than the row below it. How many bricks should be ordered if the wall is to be 20 rows tall? 10-2 NAME Answers (Lesson 10-2) PERIOD Writing Figurative Numbers as Finite Arithmetic Series Enrichment DATE A6 Glencoe Precalculus 005_036_PCCRMC10_893811.indd 14 Chapter 10 14 Glencoe Precalculus figures increased by 1, so did the factor multiplied by k and the number subtracted from that product. k=1 18, 34, 55; I used the rule ∑ (5k - 4), as the number of sides in the n 6. Study the pattern in the sigma notations for your answers. Use it to predict the first 5 heptagonal (seven-sided) numbers. Explain your reasoning. Sample answer: 1, 7, ∑ (4k - 3); The second hexagonal number has 5 more dots than the k=1 first; the third has 9 more than the second; the fourth has 13 more than the third and so forth. The number of extra dots forms the arithmetic sequence 5, 9, 13, … . n 5. The first five hexagonal numbers are 1, 6, 15, 28, and 45. Write an expression using sigma to represent the nth hexagonal number. Explain your reasoning. ∑ (3k - 2); The second pentagonal number has 4 more dots than the k=1 first; the third has 7 more than the second; the fourth has 10 more than the third and so forth. The number of extra dots forms the arithmetic sequence 4, 7, 10, … . n 4. Write an expression using sigma to represent the nth pentagonal number. Explain your reasoning. 3. The first five pentagonal numbers are 1, 5, 12, 22, and 35. Use dots to show why 12 is the third pentagonal number. Pentagonal numbers can be represented by a regular pentagon using that same number of dots. ∑ (2k - 1); The second square number has 3 more dots than the first; k=1 the third has 5 more than the second; the fourth has 7 more than the third and so forth. The number of extra dots forms the arithmetic sequence 3, 5, 7, … . n 2. Write an expression using sigma to represent the nth square number. Explain your reasoning. Likewise, square numbers are numbers that can be represented by a square using that same number of dots. ∑ k; The second triangular number has 2 more dots than the first; k=1 the third has 3 more than the second; the fourth has 4 more than the third and so forth. The number of extra dots forms the arithmetic sequence 2, 3, 4, … . n 1. Write an expression using sigma to represent the nth triangular number. Explain your reasoning. (Hint: Consider the number of extra dots needed to make the next triangular number from the previous triangular number.) 3/20/09 6:49:17 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 10 Triangular numbers are numbers that can be represented by a triangle using that same number of dots. The first three triangular numbers are 1, 3, and 6. 10-2 NAME DATE PERIOD Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 15 Chapter 10 7, 21, 63, 189, 567 15 Lesson 10-3 12/4/09 4:26:34 PM Glencoe Precalculus 6. Write a geometric sequence that has three arithmetic means between 7 and 567. 5. Find r of the geometric sequence for which a1 = 15 and a10 = 7680. 2 4. Find the first term of the geometric sequence for which a6 = 0.1 and r = 0.2. 312.5 3. Find the 17th term of the geometric sequence 128, 64, 32, … . about 0.00195 2. Find the seventh term of the geometric sequence 157, -47.1, 14.13, … . 0.114453 1. Determine the common ratio and find the next three terms of the geometric sequence x, 2x, 4x, … . r = 2, 8x, 16x, 32x Exercises Determine the geometric means recursively. a2 = 6(3) or 18, a3 = 18(3) or 54 The sequence is 6, 18, 54, 162. Write a sequence that has two geometric means between 6 and 162. Example 2 ? _____, ? 162. The sequence will resemble 6, _____, This means that n = 4, a1 = 6, and a4 = 162. Find r. Formula for nth term of a geometric sequence an = a1r n - 1 162 = 6r3 Substitute. 27 = r3 Divide each side by 6. 3=r Take the cube root of each side. Find the seventh term of the geometric sequence 8, -24, 72, … . Example 1 First, find the common ratio. a2 ÷ a1 = -24 ÷ 8 or -3 a3 ÷ a2 = 72 ÷ (-24) or -3 Use the explicit formula an = a1(r) n - 1 to find a7. Use n = 7, a1 = 8, and r = -3. a7 = 8 (-3)7 - 1 = 5832 A geometric sequence is a sequence in which each term after the first, a1, is the product of the preceding term and the common ratio, r. Therefore, to find the common ratio, divide any term by the previous term. Any nth term can be calculated with the formula an = a1r n -1. The terms between two nonconsecutive terms of a geometric sequence are called geometric means. Geometric Sequences and Series Study Guide and Intervention Geometric Sequences 10-3 NAME Answers (Lesson 10-2 and Lesson 10-3) Chapter 10 DATE Geometric Sequences and Series Study Guide and Intervention PERIOD a -ar ) ≈ 325.246 12 A7 n-1 . about 74.858 (8) n-1 20.8 ∞ 16 n=1 Glencoe Precalculus Answers Glencoe Precalculus 6. ∑ 3n - 1 does not exist 42.227 3/20/09 6:49:27 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 16 Chapter 10 n=1 3 5. ∑ 13 − ∞ If possible, find the sum of each infinite geometric series. 4. Find the sum of the first 16 terms in a geometric series where a1 = 1, and an = -2an - 1. -21,845 n=1 3. Find ∑ 5 (1.06) 11 9 2. Find the sum of a geometric series if a1 = 8, and an = 0.394, and r = − . 11 1. Find the sum of the first seven terms of -1 + (-4) + (-16) + … . -5461 Exercises 1 - 0.2 40 =− or 50 1-r S=− a1 The common ratio is 8 ÷ 40 or 0.2. Because |0.2| < 1, the series has a sum. Example 2 If possible, find the sum of the geometric series 40 + 8 + 1.6 + … . S12 1 - rn terms is known, use Sn = a1 − . Substitute 12 for n, 6 for a1, and 1.25 for r. 1-r ( 1 - 1.25 = 6( − ) 1 - 1.25 Example 1 Find the sum of the first 12 terms of the geometric series 6 + 7.5 + 9.375 + … . The common ratio is 7.5 ÷ 6 or 1.25. Because the first term and number of 1-r An infinite geometric series converges if |r| < 1 and its sum is given by S = − . ( 1-r 1 - rn If you know the first term and the number of terms, a1 and n, use Sn = a1 − . 1-r a1 1 n If you know the first and last terms, a1 and an, use the formula Sn = − . ) (continued) A geometric series is the sum of the terms of a geometric sequence. You can use a formula to find the sum of a finite geometric series or the partial sum of an infinite geometric series. Geometric Series 10-3 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Geometric Sequences and Series Practice DATE PERIOD 4 16 64 256 3 81 243 27 − ; -− , -− , -− 4 9 2. -4, -3, - − ,… n-1 81 √ 3 4 20 100 ≈ -336.99 n=2 14. ∑ 3 (0.2)n-1 0.74976 6 629,145 ∞ 3 n=2 (3) 1 16. ∑ 6 − n-1 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 17 Chapter 10 17 Lesson 10-3 12/4/09 2:48:18 PM Glencoe Precalculus 12. a1 = -3, an = 786,432, r = -4 16, -8, 4, or -16, -8, -4 17. POPULATION A city of 100,000 people is growing at a rate of 5.2% per year. Assuming this growth rate remains constant, estimate the population of the city five years from now. about 128,848 20 15. 10 + 5 + 2.5 + … 2 10. -32 and -2; 3 means If possible, find the sum of each infinite geometric series. n=3 13. ∑ -2 (1.5) 11 ≈1.84351 3 9 27 11. first eight terms of − +− +− +… Find each sum. 1, 0.5 9. 2 and 0.25; 2 means 32 , -3, 3 √3 , … 8. a9 for √3 1 − 4 1 1 6. a3 for a6 = − ,r=− Find the indicated geometric means for each pair of nonconsecutive terms. 2 7 − 7. a1 for a4 = 28, r = 2 0.0000002 5. a5 for 20, 0.2, 0.002, … Find the specified term for each geometric sequence or sequence with the given characteristics. ( 2) 3 a1 = 12; an = - − an - 1 a1 = 2; an = (5) an - 1 n-1 3 an = 12 - − ( 2) 4. 12, –18, 27, … an = 2 (5)n - 1 3. 2, 10, 50, … Write an explicit formula and a recursive formula for the nthterm of each geometric sequence. -2; 8, -16, 32 1. -1, 2, -4, … Determine the common ratio and find the next three terms of each geometric sequence. 10-3 NAME Answers (Lesson 10-3) A8 Glencoe Precalculus 005_036_PCCRMC10_893811.indd 18 Chapter 10 $414,113 b. To the nearest dollar, how much did he earn for his first 10 years of work? $33,700 a. What was his pay for his first year of work? 4. SALARY An employee agreed to a salary plan where his annual salary increases by 4.5% each year. He earned $50,081.41 for his tenth year of work. 30,105,252 Source: U.S. Census Bureau 3. POPULATION From 1990 to 2000, Florida’s population grew by about 23.5%. The population in the 2000 census was 15,982,378. If this rate of growth continues, what will be the approximate population in 2030? about 1.75 million bacteria 2. BACTERIA A colony of bacteria grows at a rate of 10% per day. If there were 100,000 bacteria on a surface initially, about how many bacteria would there be after 30 days? $6688.15 18 PERIOD 0 250 Day Amount (mg) 125 5 1 2 3 15 4 20 62.5 31.25 15.625 10 (2) ; n-1 0 20 40 60 80 100 120 140 y 2 4 6 10x Glencoe Precalculus 8 d. Graph the function that represents the sequence. 0.24 mg c. How much Bismuth-210 will the scientist have after 50 days? Round to the nearest hundredth. 1 an - 1 a1 = 125, an = − 1 an = 125 − (2) b. The amounts of Bismuth-210 can be written as a sequence with the half-life number as the domain. Write an explicit and recursive formula for finding the nth term of the geometric sequence. 0 Half-Life a. Complete the table to show the amount of Bismuth-210 every five days. 5. SCIENCE Bismuth-210 has a half-life of 5 days. This means that half of the original amount of the substance decays every five days. Suppose a scientist has 250 milligrams of Bismuth-210. Geometric Sequences and Series Word Problem Practice DATE 3/20/09 6:49:34 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 10 1. ACCOUNTING Each year, the value of a car depreciates by 18%. If you bought a $22,000 car in 2009, what will be its value in 2015? 10-3 NAME Enrichment DATE PERIOD A(1 + I ) (1 + I)n - 1 I ] [ ] , 1 - (1 + I) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 19 Chapter 10 $143.53 3. $1200 at 18% per year for nine months $166.98 1. $6000 at 15% per year for four years 19 $241.52 . Lesson 10-3 3/24/09 3:49:39 PM Glencoe Precalculus 4. $11,500 at 9.5% per year for five years $829.65 2. $75,000 at 13% per year for thirty years Find the amount of the monthly payment for each loan. If payments are made monthly, then I is the monthly interest rate and n is the total number of monthly payments. AI . which simplifies to a formula for determining the monthly payment, p = − -n [ 0 = A(1 + I)n - p − . Solving for p gives p = (1 + I ) - 1 − I − n n 1 - (1 + I) -I 1+I = − and r = (1 + I)n - 1 or − for the expression in brackets. Since the last balance bn equals 0, I n The formula for the sum of a geometric series gives Sn = 1 − 1 - 1(1 + I )n 1 - (1 + I ) Continue this pattern to the nth balance. bn = A(1 + I)n - p[(1 + I)n - 1 + (1 + I)n - 2 + . . . + (1 + I) + 1] The expression in brackets is a geometric series with a = 3 2 b3 = b2 + b2I - p = b2(1 + I) - p or A(1 + I ) - p(1 + I ) - p(1 + I ) - p b2 = b1 + b1I - p = b1(1 + I) - p or A(1 + I)2 - p(1 + I) - p b1 = A + AI - p = A(1 + I) - p Fill in the blanks. Many installment loans, including home mortgages, credit card purchases, and some car loans, compute interest only on the outstanding balance. Part of each equal payment goes for interest and the remainder reduces the amount owed. As the outstanding balance decreases, the amount of interest paid each term decreases. Let A represent the amount borrowed, p the amount of each payment, I the interest rate, n the number of payments, and bk the balance after k payments. The first balance b1 equals the amount borrowed A plus the interest A(I) minus one payment p. The second balance b2 equals b1 plus the interest b1(I) minus another payment p and so on. Installment Loans 10-3 NAME Answers (Lesson 10-3) Chapter 10 TI-Nspire Activity DATE Find the 15th term of the sequence an = 4 (2)n - 1. PERIOD A9 20 n-1 . 32 Glencoe Precalculus Answers Glencoe Precalculus 12/4/09 2:48:55 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 20 Chapter 10 1 7. Find the sum of the first 30 terms of the sequence an = 24 − (4) 6. Find the sum of the first 15 terms of the sequence in Exercise 3. 999.969 5. Find the sum of the first 10 terms of the sequence in Exercise 2. 5115 4. Find the sum of the first 7 terms of the sequence in Exercise 1. 3825.5 3. Find the 15th term of the sequence an = 500 (0.5)n - 1. 0.030518 2. Find the 10th term of the sequence an = 5(2)n - 1. 2560 1. Find the 7th term of the sequence an = 3.5(3)n - 1. 2551.5 Exercises Method 2: Insert a CALCULATOR page. Press the catalog key and choose ∑ (. As shown at the bottom of the Catalog page, the format is (expression, variable, low, high). Type 4(2)n - 1, n, 1, 15) and press ·. Method 1: These are the terms in column A. Move to cell B1 and type the formula = SUM(A1 : A15) and press ·. The sum is 131,068. Example 2 Find the sum of the first 15 terms of the sequence in Example 1. You can find the sum of the first n terms in either the LISTS & SPREADSHEET application or in the CALCULATOR application. Add a LISTS & SPREADSHEET page. Move the cursor to the gray cell above A1 and select MENU > DATA > GENERATE SEQUENCE. Type the formula 4(2)n - 1 next to u(n) =, enter 4 as the initial term, and enter 15 as the maximum number of terms. Be sure to use parentheses to indicate the exponent. Tab down to OK and press ·. Column A is now populated with the first 15 terms of the sequence. The 15th term is 65,536. Example 1 You can generate terms of a sequence in the LISTS & SPREADSHEET application. Finding Terms and Sums 10-3 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. DATE PERIOD (k + 1) is a common factor. Add 2(k + 1) to both sides. Replace n with k. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 21 Chapter 10 21 Lesson 10-4 12/4/09 2:56:00 PM Glencoe Precalculus 1 + 5 + 9 + … + (4k − 3) = k (2k − 1) 1 + 5 + 9 + … + (4k − 3) + 4k + 1 = k (2k − 1) + 4k + 1 = 2k 2 + 3k + 1 = (k + 1) (2k + 1) = (k + 1) (2 (k + 1) − 1) The conjecture is true for k + 1 when it is true for k. Therefore, it is true for all positive integers n. Verify the statement is valid for n = 1: 1(2(1) -1) = 1. Assume the statement is true for n = k, and prove that for k + 1, Pk + 1 = (k + 1) (2 (k + 1) − 1). 1. Use mathematical induction to prove that 1 + 5 + 9 + 13 + . . . + (4n - 3) = n(2n - 1) is true for all positive integers n. Exercise Thus, if the formula is true for n = k, it is also true for n = k + 1. Since Pn is true for n = 1, it is also true for n = 2, n = 3, and so on. That is, the formula for the sum of the first n positive even integers is true for all positive integers n. (k + 1)[(k + 1) + 1] or (k + 1)(k + 2) If k + 1 is substituted into the original formula (n(n + 1)), the same result is obtained. Pk + 1 ⇒ 2 + 4 + 6 + . . . + 2k + 2(k + 1) = (k + 1)(k + 2) We can simplify the right side by adding k(k + 1) + 2(k + 1). Pk + 1 ⇒ 2 + 4 + 6 + . . . + 2k + 2(k + 1) = k(k + 1) + 2(k + 1) Next, prove that Pn is also true for n = k + 1. Pk ⇒ 2 + 4 + 6 + . . . + 2k = k(k + 1). Step 2: Then assume that Pn is true for n = k. Step 1: First, verify that P n is true for the first possible case, n = 1. Since the first positive even integer is 2 and 1(1 + 1) = 2, the formula is true for n = 1. Here P n is defined as 2 + 4 + 6 + . . . + 2n = n(n + 1). Example Use mathematical induction to prove that the sum of the first n positive even integers is n(n + 1). A method of proof called mathematical induction can be used to prove certain conjectures and formulas. A conjecture can be proven true if you can show that something works for the first case, assume that it works for any particular case, and then show it works for the next case. Mathematical Induction Study Guide and Intervention Mathematical Induction 10-4 NAME Answers (Lesson 10-3 and Lesson 10-4) Mathematical Induction Study Guide and Intervention DATE Prove that n! > 5 for integer values of n ≥ 12. (k + 1)! > (k + 1) · 5 Definition of factorial Multiply each side by k + 1. Inductive hypothesis A10 Transitive Property of Inequality Simplify using a property of exponents. (k + 1)! > 5 · 5 k (k + 1)! > 5 k + 1 Simplify. Multiply each side by 4. Inductive hypothesis Glencoe Precalculus 005_036_PCCRMC10_893811.indd 22 Chapter 10 22 Glencoe Precalculus 16k - 4(k + 1) = 12k - 4. Because k > 2, 12k - 4 > 0 and 16k - 4(k + 1) > 0. By the Addition Property of Inequality, 16k > 4(k + 1). Combining inequalities, we have 4 k + 1 > 16k > 4(k + 1). By the Transitive Property of Inequality, we have 4 k + 1 > 4 (k + 1). Because P n is valid for n = 2 and for n = k + 1, it is valid for all integers n ≥ 2. 4 k + 1 > 16k 4 · 4 k > 4 · 4k 4 k > 4k Assume that 4 k > 4k is true for some integer k > 2. Show that 4 k + 1 > 4(k + 1) is true. Let P n be the statement that 4 n > 4n for integer values n ≥ 2. When n = 2, 4 2 > 4(2) or 16 > 8 is true. Therefore, it is true for the first possible case. 1. Prove that 4 n > 4n for n ≥ 2. Exercise Because P n is true for n = 12 and for n = k + 1, it is true for all integers n ≥ 12. Therefore, (k + 1)! > 5 k + 1 is true. Combined inequality (k + 1)! > (k + 1) · 5 k > 5 · 5 k For k > 12, we know that k + 1 > 5. The Multiplication Property of Inequality states we can multiply each side of an inequality by a positive value and maintain the inequality. Therefore, we can multiply each side of k + 1 > 5 by 5 k to obtain (k + 1) · 5 k > 5 · 5 k. k (k + 1) · k! > (k + 1) · 5 k k! > 5 k Step 2: Assume P n is true for n = k, so assume k! > 5 k for some positive integer k > 12. Show that (k + 1)! > 5 k + 1 is true. 12! = 479,001,600 and 5 12 = 244,140,625 and 479,001,600 > 244,140,625. Step 1: Let P n be the statement that n! > 5 n for integer values n ≥ 12. The first possible case is n = 12. Verify that P n is true for n = 12. Example n The extended principle of mathematical induction is used when a statement is not true for n = 1. The first step is to prove that P n is true for the first possible case. (continued) PERIOD 12/4/09 2:58:40 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 10 Extended Mathematical Induction 10-4 NAME Mathematical Induction Practice DATE PERIOD 3 3 3 6 3 3 3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 23 Chapter 10 23 Lesson 10-4 12/4/09 4:51:24 PM Glencoe Precalculus Step 1: Verify that P n is valid for n = 1. P 1 ⇒ 5 1 + 3 = 8. Since 8 is divisible by 4, P n is valid for n = 1. Step 2: Assume that P n is valid for n = k and then prove that it is valid for n = k + 1. P k ⇒ 5k + 3 = 4r for some integer r Pk + 1 ⇒ 5k + 1 + 3 = 4t for some integer t 5k + 3 = 4r 5(5 k + 3) = 5(4r) 5 k + 1 + 15 = 20r 5 k +1 + 3 = 20r - 12 5 k+ 1 + 3 = 4(5r - 3) Let t = 5r - 3, an integer. Then 5 k - 1 + 3 = 4t. Thus, if P k is valid, then Pk + 1 is also valid. Since P n is valid for n = 1, it is also valid for n = 2, n = 3, and so on. Hence, 5 n + 3 is divisible by 4 for all positive integers n. 2. 5 n + 3 is divisible by 4. Thus, if the formula is valid for n = k, it is also valid for n = k + 1. Since the formula is valid for n = 1, it is also valid for n = 2, n = 3, and so on. That is, the formula is valid for all positive integers n. (k + 1)(k + 1 + 1) (k + 1)(k + 2) − or − 6 6 Apply the original formula for n = k + 1. Pk + 1 3 1 2 3 k Pk ⇒ − +−+−+…+− = − k(k + 1) 6 k(k + 1) k+1 1 2 3 k k+1 ⇒− +−+−+…+− + −= − + − 3 3 3 3 3 6 3 k(k + 1) + 2(k + 1) = − 6 (k + 1) (k + 2) = − 6 1 Step 1: Verify that the formula is valid for n = 1. Since − 3 1(1 + 1) 1 is the first term in the sentence and − = −, the formula 6 3 is valid for n = 1. Step 2: Assume that the formula is valid for n = k and then prove that it is also valid for n = k + 1. 3 n(n + 1) 3 n 1 2 1. − +− +− + ... + − =− Use mathematical induction to prove that each conjecture is valid for all positive integers n. 10-4 NAME Answers (Lesson 10-4) Chapter 10 3 A11 24 Glencoe Precalculus Answers Glencoe Precalculus In both cases, P n is valid for n = k + 1. Because P n is valid for n = 6 and n = k + 1, it is valid for all n that are integers greater than or equal to 6. All gravel sales greater than 25 pounds can be loaded using the 10- and 25-pound buckets. Case 2: P k contains no 25-pound buckets. P k must contain at least three 10-pound buckets. Replace two of these buckets with a 25-pound bucket, and the value of P k is increased by 5 to 5k + 5 or 5(k + 1), which is P k + 1. Case 1: P k contains at least one 25-pound bucket. Replace one 25-pound bucket with 3 10-pound buckets, and the value of P k is increased by 5 to 5k + 5 or 5(k + 1), which is P k + 1. 12/9/09 5:43:06 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 24 Chapter 10 So, when the statement is true for k, it is true for k + 1. k(k − 3) k(k − 3) + 2(k − 1) + (k − 1) = − − 2 2 k 2 − 3k + 2k −2 = − 2 k2 − k − 2 = − 2 (k + 1)(k − 2) = − 2 (k + 1) ((k + 1) − 3) = −. 2 polygon with k + 1 sides equals number of diagonals in a convex k ≥ 3 sides is − . Then the k(k − 3) 2 Assume that the number of diagonals in a convex polygon with with 0 diagonals. Also, − = 0. 0(0 − 3) 2 If n = 3, then the figure is a triangle c. Use the extended principle of mathematical induction to prove the statement above. When a vertex is added, one of the sides becomes a diagonal. Also, diagonals are drawn from that point to every other point except the two consecutive vertices. This is 1 + n - 2, which simplifies to n - 1. b. Explain why for every additional vertex added to the polygon, the number of diagonals increases by n - 1. Assume that for n = k, there exists a set of 10- and/or 25 pound buckets that adds to 5k. Show that P n is valid for n = k + 1. There are two cases to consider. Let P n be the set of 10-pound and 25-pound bucketfuls of gravel that add to 5n for n > 5. For n = 6, 30 pounds of gravel the conjecture is valid because 10(3) = 30. 2 The number of diagonals in a convex n(n − 3) polygon with n sides is equal to − . a. What is the least possible value of n? PERIOD 2. GRAVEL The gravel at a stone center is sold in 5-pound increments. Customers can load their trucks by using either 10-pound or 25-pound buckets. Prove that all gravel sales greater than 25 pounds can be loaded using just the 10- and 25-pound buckets. Mathematical Induction Word Problem Practice DATE 1. GEOMETRY Diagonals are segments that join nonconsecutive vertices. 10-4 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Enrichment DATE Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 25 Chapter 10 25 Verify that S n is true for n = 1. S 1 = 5(1) - 4 or 1. Assume S n is true for n = k. Prove it is true for n = k + 1. c. Find a formula that can be used to compute the number of dots in the nth diagram of this sequence. Use mathematical induction to prove your formula is correct. S n = 5n - 4 Add 5 to the number of dots in S 4. S 5 would have 16 + 5 or 21 dots. b. Describe a method that you can use to determine the number of dots in the fifth diagram S 5 based on the number of dots in the fourth diagram, S 4. Verify your answer by constructing the fifth diagram. 16 a. How many dots would there be in the fourth diagram S4 in the sequence? 2. Refer to the diagrams at the right. n = 5: 5 2 = 25, 32 = 2 5, 25 < 32 Assume the statement is true for n = k. Prove it is true for n = k + 1. (k + 1) 2 = k 2 + 2k + 1 < k 2 + (k - 1)k + 1 < k2 + 2k + 1 - k < 2k + 2k < 2k + 1 So, the statement is true for n > 4. 0 PERIOD f(x) 4 4T x Lesson 10-4 12/4/09 3:10:09 PM Glencoe Precalculus S k + 1 = Sk + 5 S k + 1 = 5(k + 1) - 4 = 5k + 5 - 4 = (5k - 4) + 5 = Sk + 5 4 4 f(x) = x 2 g(x) = 2 x since 2 < k - 1 since k 2 < 2 k since 1 - k < 0 since 2 k + 2 k = 2 k + 1 c. Use mathematical induction to prove your response from part b. b. Write a conjecture that compares n 2 and 2 n, where n is a positive integer. If n > 4, n 2 < 2 n. 1. a. Graph f(x) = x 2 and g(x) = 2 x on the axes shown at the right. Frequently, the pattern in a set of numbers is not immediately evident. Once you make a conjecture about a pattern, you can use mathematical induction to prove your conjecture. Conjectures and Mathematical Induction 10-4 NAME Answers (Lesson 10-4) The Binomial Theorem Study Guide and Intervention DATE 1 1 4 1 3 1 6 2 1 3 1 4 1 1 Use Pascal’s triangle to expand (x + 2y) 5. A12 Glencoe Precalculus 7 6 5 2 4 3 3 4 2 5 6 26 Glencoe Precalculus c + 7c d + 21c d + 35c d + 35c d + 21c d + 7cd + d 005_036_PCCRMC10_893811.indd 26 Chapter 10 6. (c + d) 7 5. (2a – 2b)5 32a5 - 160a4b + 320a3b2 - 320a2b3 + 160ab4 - 32b5 4. (m - n)6 m6 - 6m5n + 15m4n2 - 20m3n3 + 15m2n4 - 6mn5 + n6 7 81x4 + 108x3y + 54x2y2 + 12xy3 + y4 2. (3x + y)4 3. (7 + g)4 2401 + 1372g + 294g2 + 28g3 + g4 x3 + 12x2 + 48x + 64 1. (x + 4)3 Use Pascal’s triangle to expand each binomial. Exercises The numbers in the fifth row of Pascal’s triangle are the coefficients. Following the pattern above, these numbers will be 1, 5, 10, 10, 5, and 1. Add coefficients. 1x5 + 5x4 (2y) + 10x3 (4y2) + 10x2 (8y3) + 5x (16y4) + 1 · 32y5 x5 + 10x4y + 40x3y2 + 80x2y3 + 80xy4 + 32y5 Simplify. First, write the series for (a + b)5 without coefficients. Then replace a with x and b with 2y. Series for (a + b)5 a5b0 + a4b1 + a3b2 + a2b3 + a1b4 + a0b5 x5 (2y)0 + x4 (2y)1 + x3 (2y)2 + x2 (2y)3 + x1 (2y)4 + x0 (2y)5 Substitution x5 + x4 (2y) + x3 (4y2) + x2 (8y3) + x (16y4) + 32y5 Simplify. Example The numbers in Pascal’s triangles are the binomial coefficients when (a + b)n is expanded. You can use these numbers to expand binomials without multiplying repeatedly. The first term is an, the last term is bn, and the powers of a decrease by 1 as the powers of b increase by 1 from left to right. Row 4 Row 3 Row 2 Row 1 Row 0 1 PERIOD 12/4/09 3:09:49 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 10 Pascal’s Triangle In Pascal’s triangle, the first and last numbers in each row is 1 and the number in row 0 is 1. Other numbers are the sum of the two numbers above them. The first five rows of Pascal’s triangle are shown below. 10-5 NAME The Binomial Theorem Study Guide and Intervention DATE (continued) PERIOD (6 - 3)!3! 3!3! 3!3! 1,548,288 2. (3a + 4b)8, a3b5 term 3 2 2 3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 27 Chapter 10 x - 8x y + 24x y - 32xy + 16y 4 5. (x - 2y)4 x5 + 15x4 + 90x3 + 270x2 + 405x + 243 3. (x + 3)5 4 27 Lesson 10-5 12/4/09 3:09:29 PM Glencoe Precalculus 16x4 - 96x3y + 216x2y2 - 216xy3 + 81y4 6. (2x - 3y)4 64x3 + 96x2y + 48xy2 + 8y3 4. (4x + 2y)3 Use the Binomial Theorem to expand each binomial. 2500 1. (x + 5)6, fourth term Find the coefficient of the indicated term in each expansion. Exercises Example 2 Use the Binomial Theorem to expand (3x + 7)4. Let a = 3x and b = 7. (3x + 7)4 = 4C0(3x)4(7)0 + 4C1(3x)3(7)1 + 4C2(3x)2(7)2 + 4C3(3x)1(7)3 + 4C4(3x)0(7)4 = 1 · 81x4 · 1 + 4 · 27x3 · 7 + 6 · 9x2 · 49 + 4 · 3x · 343 + 1 · 1 · 2401 = 81x4 + 756x3 + 2646x2 + 4116x + 2401 The binomial coefficient of the fourth term in (a + b)6 is 20. Substitute for a and b in an - rbr. 20(5a)6 – 3(2b)3 = 20(5a)3(2b)3 = 20(125a3)(8b) = 20,000a3b The coefficient is 20,000. 6 6! 6! · 5 · 4 · 3! C3 = − =− = 6− or 20 Example 1 Find the coefficient of the fourth term in the expansion of (5a + 2b)6. For (5a + 2b)6 to have the form (a + b)n, let a = 5a and b = 2b. Since r increases from 0 to n, r is one less than the number of the term. Evaluate 6C3. The Binomial Theorem states that for any positive integer n, the expansion of (a + b)n is C anb0 + nC1 an - 1b1 + nC2 an - 2b2 + … + nCr an - rbr + … + nCn a0bn. n 0 (n - r)! r! n! . calculator or by finding − The Binomial Theorem The binomial coefficient of the an - r br term in the expansion of (a + b)n is given by nCr. You can find nCr by using a 10-5 NAME Answers (Lesson 10-5) Chapter 10 The Binomial Theorem Practice A13 5 180 64p6 - 576p5q + 2160p4q2 - 4320p3q3 + 4860p2q4 - 2916pq5 + 729q6 10. (2p - 3q)6 81x4 + 216x3y + 216x2y2 + 96xy3 + 16y4 8. (3x + 2y)4 ( ) 28 Glencoe Precalculus Answers Glencoe Precalculus 12/5/09 3:25:04 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 28 Chapter 10 256 12. SPORTS A varsity volleyball team needs nine members. Of these nine members, at least five must be seniors. How many of the possible groups of juniors and seniors have at least five seniors? 15 ∑ (3x)15 - r(8y)r r r=0 15 11. Represent the expansion of (3x + 8y)15 using sigma notation. a5 - 5 √ 2 a4 + 20a3 - 20 √ 2 a2 + 20a - 4 √ 2 9. (a - √ 2) x4 - 20x3 + 150x2 - 500x + 625 7. (x - 5)4 6 6. (a - 2 √ 3 ) , 3rd term 286,720 4. (4a + 2b)8, 5th term Use the Binomial Theorem to expand each binomial. 10,206 5. (3p + q)9, q5p4 term -216 3. (2n - 3m)4, 4th term PERIOD 81a4 + 108a3b + 54a2b2 + 12ab3 + b4 2. (3a + b)4 DATE Find the coefficient of the indicated term in each expansion. r5 + 15r4 + 90r3 + 270r2 + 405r + 243 1. (r + 3)5 Use Pascal’s triangle to expand each binomial. 10-5 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 1.6% 9.4% 23.4% 31.3% 23.4% 9.4% 1.6% 0 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 29 Chapter 10 0.8% 3. PROMOTION A juice company is holding a promotion where one in every five bottles of juice has a coupon for a free bottle of juice. If a customer buys three bottles, what is the probability that all three bottles have a free juice coupon? 34.4% b. What is the probability that at least four of the children are girls? 6 5 4 3 2 1 Probability Number of Girls a. Complete the table to show the probability that they have each number of girls. 2. FAMILY Suppose a mother and father have 6 children. Assume that having a girl or boy are equally likely outcomes. 16.8% 29 The Binomial Theorem DATE PERIOD 8.9% Lesson 10-5 3/20/09 6:50:26 PM Glencoe Precalculus 6. WORK The probability that a substitute teacher has to work on a Friday during any given week in a certain school district is 32%. What is the probability that the substitute teacher will work on three of the four Fridays in the upcoming month? 3.2% c. What is the probability that all 75 apples go into a pie? 56.2% b. What is the probability that at least 72 of the apples go into a pie? 87.8 % a. What is the probability that 5 or fewer of the apples will be rejected? 5. FOOD The probability that an apple does not meet the quality-control standards for continuing down an assembly line to become filling for a pie is 4.5%. A batch of 75 apples is received. 72.5% 4. SPORTS On average, a basketball player misses 3 free throws out of every 8 attempts. If the player attempts 5 free throws, what is the probability that he misses no more than two times? Word Problem Practice 1. GOLF A golfer can drive a ball to the fairway about 70% of the time. What is the probability of hitting the fairway on exactly 14 of the 18 holes? 10-5 NAME Answers (Lesson 10-5) Enrichment PERIOD A14 Glencoe Precalculus 005_036_PCCRMC10_893811.indd 30 Chapter 10 30 Glencoe Precalculus In any row of Pascal’s triangle after the first, the sum of the odd numbered terms is equal to the sum of the even numbered terms. d. Repeat parts a through c for at least three other rows of Pascal’s triangle. What generalization seems to be true? The sums are equal. c. How do the sums in parts a and b compare? See students’ work. b. In the same row, find the sum of the even numbered terms. See students’ work. a. Starting at the left end of the row, find the sum of the odd numbered terms. 2. Pick any row of Pascal’s triangle that comes after the first. Sum of numbers in row n = 2n - 1; The sum of the numbers in the rows above row n is 20 + 21 + 22 + . . . + 2n - 2, which, by the formula for the sum of a geometric series, is 2n - 1 - 1. e. See if you can prove your generalization. It appears that the sum of the numbers in any row is 1 more than the sum of the numbers in all of the rows above it. d. Repeat parts a through c for at least three more rows of Pascal’s triangle. What generalization seems to be true? The answer for part b is 1 more than the answer for part a. c. How are your answers for parts a and b related? See students’ work. b. What is the sum of all the numbers in the row that you picked? See students’ work. a. What is the sum of all the numbers in all the rows above the row that you picked? 1. Pick a row of Pascal’s triangle. You have learned that the coefficients in the expansion of (x + y)n yield a number pyramid called Pascal’s triangle. This activity explores some of the interesting properties of this famous number pyramid. DATE 3/20/09 6:50:29 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 10 Patterns in Pascal’s Triangle 10-5 NAME DATE 1-x n=0 ∞ PERIOD ∞ n=0 Use ∑ xn to find a power series representation of ∞ 1 - 3x 2-x [-0.5, 0.5] scl: 0.1 by [-1, 6] scl: 1 2-x Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 31 Chapter 10 n=0 31 1 g(x) = − = ∑ (x - 1)n for 0 < x < 2 ∞ calculator to graph g(x) together with the sixth partial sum of its power series. Lesson 10-6 10-1 12/4/09 3:19:49 PM Glencoe Precalculus [-0.5, 2.5] scl: 1 by [-1, 5] scl: 1 Indicate the interval on which the series converges. Use a graphing n=0 1 1. Use ∑ x n to find a power series representation of g(x) = − . ∞ Exercise 1 and S6(x) are shown. The graphs of g(x) = − n=0 ∑ (3x)n or 1 + 3x + (3x)2 + (3x)3 + (3x)4 + (3x)5 5 1 1 The series converges for |3x| < 1, which can be written as - − <x<− . 3 3 Find the sixth partial sum. n=0 Replace x with 3x in f(x) to get f(3x) = ∑ (3x)n for |3x| < 1. Here, g(x) = f(3x). transformation, write g(x) = f(u) and solve for u. n=0 f(x) = ∑ xn for |x| < 1 and g(x) is a transformation of f(x). To find the ∞ Use a graphing calculator to graph g(x) together with the sixth partial sum of its power series. 1 - 3x 1 . Indicate the interval on which the series converges. g(x) = − Example a2x2 + a3x3 + …, where x and a can take on any values n = 0, 1, 2, … . n=0 A power series in x is an infinite series of the form ∑ anxn = a0 + a1x + for|x| < 1. can be expressed as the infinite series ∑ xn or 1 + x + x2 + … + x n ∞ 1 The rational function f(x) = − Functions as Infinite Series Study Guide and Intervention Power Series 10-6 NAME Answers (Lesson 10-5 and Lesson 10-6) Chapter 10 Functions as Infinite Series Study Guide and Intervention DATE (continued) PERIOD n! 2! 3! 4! 5! ∞ (-1)nx2n (2n)! 4 4! 2 2! 6 8 6! 8! (-1)nx2n + 1 x3 x5 x7 x9 sin x = ∑ − = x - − +− -− +− -… 3! 5! 7! 9! (2n + 1)! n=0 n=0 ∞ x x x x +− -− +− -… cos x = ∑ − = 1 - − n=0 xn x2 x3 x4 x5 ex = ∑ − =1+x+− +− +− +− +… 5 5! 3! 7 9 A15 9! π 4 -i− iπ − 4e 6 32 Glencoe Precalculus Answers Glencoe Precalculus 12/4/09 3:20:13 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 32 Chapter 10 8 4. Use the fifth partial sum of the trigonometric series for cosine to π approximate the value of cos − . Round to three decimal places. ≈0.924 3. Write 2 √ 3 + 2i in exponential form. 2. Use the fifth partial sum of the exponential series to approximate the value of e2.7. Round to three decimal places. 12.840 1. Write 4 - 4i in exponential form. 4 √ 2e Exercises ≈ 0.500 6 (0.524)3 (0.524)5 (0.524)7 (0.524)9 π sin − ≈ 0.524 - − + − - − + − 6 3! 5! 7! 9! π Let x = − or about 0.524. 7! x x x x sin x = x - − +− -− +− -… 3 Example Use the fifth partial sum of the trigonometric series π for sine to approximate the value of sin − . Round to three decimal 6 places. iθ a2 + b2 and Exponential Form of a a + bi = re , where r = √ b b Complex Number for a > 0 and θ = tan-1 − θ = tan-1 − a a + π for a < 0 Trigonometric Series Exponential Series ∞ The value of ex can be approximated by using the exponential series. The trigonometric series can be used to approximate values of the trigonometric functions. Euler’s Formula can be used to write the exponential form of a complex number that is the natural logarithm of a negative number. Transcendental Functions as Power Series 10-6 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. ∞ Functions as Infinite Series Practice DATE 3-x n=0 [-2, 4] scl: 1 by [-1, 6] scl: 1 PERIOD 3.294 3. e 1.2 4 -0.706 − 5. cos 3π π i− 3 3 ) e 5 √2 7. 5 + 5i 4 π i− iπ + 1.7405 10. ln (-5.7) 3 -π i− iπ + 6.9078 11. ln (-1000) 2e 8. 1 - √ 3i Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 33 Chapter 10 about 15.4 years 33 Lesson 10-1 10-6 12/5/09 3:27:31 PM Glencoe Precalculus b. How long will it take for Derika’s deposit to double, provided she does not deposit any additional funds into her account? approximately $856.02 a. Approximate Derika’s savings account balance after 12 years using the first four terms of the exponential series. 12. SAVINGS Derika deposited $500 in a savings account with a 4.5% interest rate compounded continuously. (Hint: The formula for continuously compounded interest is A = Pert.) iπ + 1.386 9. ln (-4) Find the value of each natural logarithm in the complex number system. 13e 3 ( π π 6. 13 cos − + i sin − Write each complex number in exponential form. 0.501 6 5π 4. sin − Use the fifth partial sum of the trigonometric series for cosine or sine to approximate each value. Round to three decimal places. 1.648 2. e 0.5 Use the fifth partial sum of the exponential series to approximate each value. Round to three decimal places. 3-x 2 g(x) = − = ∑ [0.5(x - 1)]n for -1 < x < 3 ∞ Indicate the interval on which the series converges. Use a graphing calculator to graph g(x) together with the sixth partial sum of its power series. n=0 2 1. Use ∑ xn to find a power series representation of g(x) = − . 10-6 NAME Answers (Lesson 10-6) (0.03x)n n! A16 ) (4 ) Glencoe Precalculus 005_036_PCCRMC10_893811.indd 34 Chapter 10 original: 2.0, after 3 terms: 1.75, after 6 terms: 1.96875 b. Compare the power series approximation after 3 and 6 terms to the original equation for x = 14. n=0 n x ∑ − - 3 ; 8 < x < 16 ∞ a. Write a power series approximation for the price of this stock. Where does it converge? 4 4-− 1 P(x) = − x , where x is time in months. 3. STOCKS An analyst notices that the early growth of a stock price in hundreds of dollars per share can be modeled by 8.66 cm ( 2. MECHANICS The function π f(x) = 10 cos − x models the distance in 12 centimeters a weight on a spring is from its initial position after x seconds, without regard for friction. Use the fifth partial sum of the trigonometric series for cosine to find the distance after 2 seconds. $2323.67 b. Use the first five terms of the series to find the amount of money in the account after 5 years. n =0 P = 2000 ∑ − ∞ a. Write a power series to approximate Jill’s account balance, assuming she does not deposit any more money. 34 PERIOD 487 Pairs ∞ (0.07325x -137.67) n n! n! Glencoe Precalculus 1300; No, there are not enough terms in the sum for a good approximation. d. Use your power series to the sixth term to approximate the number of breeding pairs of bald eagles in 2012. Is this a good approximation? Why or why not? n=0 = ∑ −− n =0 ∞ n (x ln 1.076 - ln (1.620 × 10-60)) ∑ −− c. By using the logarithmic change of base formula, you can write the exponential equation with a base e. f(x) = ab x = e x ln b + ln a. Write a power series to approximate your regression equation from part a. b. Approximate the number of breeding pairs in 2012. 16,436 f(x) = 1.620 × 10 -60 (1.076) x a. Determine an exponential regression equation for this data. Use the year number for x. 791 1757 3035 4449 6471 9789 1963 1974 1984 1990 1994 2000 2006 Year 4. ENDANGERED SPECIES The bald eagle was placed on the endangered species list in 1967 and removed in 2007. The number of breeding pairs in the lower 48 states is documented below. Functions as Infinite Series Word Problem Practice DATE 12/4/09 5:05:31 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 10 1. INVESTMENT Jill deposits $2000 into an account that compounds continuously at 3.0%. 10-6 NAME Enrichment DATE PERIOD 4 9 16 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 35 Chapter 10 35 the absolute value of each term is less than or equal to the corresponding term in a p-series with p = 2. 1 1 1 1 1 − -− +− -− + … is convergent because Sample answer: 2. Create an alternating series, other than a geometric series with negative common ratio, that has a sum. Justify your answer. If the series formed by taking the absolute values of the terms of a given series is convergent, then the given series is said to be absolutely convergent. It can be shown that any absolutely convergent series is convergent. Lesson 10-6 10-1 3/20/09 6:50:53 PM Glencoe Precalculus Since 1, 0, 1, 0, … has no limit, the original series has no sum. Let S n be the nth partial sum. Then ⎧ 1 if n is odd. Sn = ⎨ ⎩ 0 if n is odd. c. Write an argument that suggests that there is no sum. (Hint: Consider the sequence of partial sums.) 1 - 1 + 1 - 1 + … = (1 - 1) + (1 - 1) + (1 - 1) + … =0+0+0+… =0 b. Write an argument that suggests that the sum is 0. 1 - 1 + 1 - 1 + … = 1 + (-1 + 1) + (-1 + 1 ) + … =1+0+0+… =1 a. Write an argument that suggests that the sum is 1. 1. Consider 1 - 1 + 1 - 1 + … . The series below is called an alternating series. 1-1+1-1+… The reason is that the signs of the terms alternate. An interesting question is whether the series converges. In the exercises, you will have an opportunity to explore this series and others like it. Alternating Series 10-6 NAME Answers (Lesson 10-6) Chapter 10 PERIOD Approximating Sine Using Polynomial Functions Graphing Calculator Activity DATE ) 4 + ( 4 Y= ) ENTER 3 ÷ 3 GRAPH . 5! ) 5 ÷ 5 MATH 3! ENTER [–9.42, 9.42] scl: 1 by [–2, 2] scl: 1 5 5! 3 7 A17 5 5! 3 3! 7 9 9! 36 Glencoe Precalculus Answers Glencoe Precalculus 12/4/09 3:24:37 PM Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 005_036_PCCRMC10_893811.indd 36 Chapter 10 11! -− 2. What term should be added to k(x) to obtain a polynomial with six terms that gives good approximations to sin x? x 11 1. Are the intervals for which you get good approximations for sin x larger or smaller for polynomials that have more terms? larger Exercises [–9.42, 9.42] scl: 1 by [–2, 2] scl: 1 7! x x x x Step 3 Repeat Step 1 using k(x) = x - − +− -− +− . [–9.42, 9.42] scl: 1 by [–2, 2] scl: 1 7! x x x Step 2 Repeat Step 1 using h(x) = x - − +− -− instead of g(x). 3! In absolute value, what are the greatest and least differences between the values of f(x) and g(x) for the values of x described by the inequality that you wrote? Step 1 Use to help you write an inequality describing the x-values for which the graphs seem very close together. Press and use and to move along the graph of y = sin x. Press or x3 x5 to move the cursor to the graph of y = x - − +− . Press and 3! 5! to move along the graph. MATH ( — same screen. Press In this activity you will examine polynomial functions that can be used to approximate sin x. x3 x5 Graph f(x) = sin x and g(x) = x - − +− on the 10-6 NAME Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Answers (Lesson 10-6) Chapter 10 Assessment Answer Key Quiz 1 (Lessons 10-1 and 10-2) Page 37 Quiz 3 (Lessons 10-4 and 10-5) Page 38 Mid-Chapter Test Page 39 30 1. 2. diverge 3. 7.875 8.57 5. 796 6. C 7. 300.3 5 + 11 + 17 + … + (6k − 1) + (6k + 5) = k(3k − 2) + 2. (6k + 5) h4 + 4h3k + 6h2k2 3 4 3. + 4hk + k Quiz 2 (Lesson 10-3) B 2. J 3. C 4. F 5. B A 1. 4. 1. 4. -13,608 5. 18.9% Quiz 4 (Lesson 10-6) 8 1. Page 38 ∞ 2. ∞ 20.78125 (2) 1 ∑ 10 − n ∑ (x - 8) ; 6. n =1 n n =1 3. a n = 2(5) n − 1 3 3 4 , 3 √ 16 , 12 4. 3, 3 √ 5. 1.5 1. 7 < x < 9 2. (1.047)3 3! (1.047)5 (1.047)7 −-− 5! 7! Chapter 10 D 7. 1.047 - − + 3. 6. 36.77 8. about 0.8659 iπ − 4. 4e 5. C 2 A18 9. 10. 7 neither, − 11 an = 3n - 7; an = an − 1 + 3, a1 = −4 11 − 120 converges Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Page 37 Chapter 10 Assessment Answer Key Form 1 Page 41 1. 2. 3. 1. 2. geometric sequence 4. explicit sequence 5. 3. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. B G D 13. A 14. H 15. A 16. H 17. C 18. G 19. B 20. J H B arithmetic series 6. 4. Page 42 H geometric means 7. 5. converge 6. binomial coefficients A 8. G 9. A 10. H 11. B Answers Vocabulary Test Page 40 series 7. 8. Fibonacci sequence 9. eiθ = cos θ + i sin θ 10. a sequence in which each term is determined by one or more of the previous terms 12. Chapter 10 J B: A19 5 − 12 Glencoe Precalculus Chapter 10 Assessment Answer Key Form 2A Page 43 Page 44 12. Form 2B Page 45 H 1. B 1. C 2. F 2. G 3. A 3. B 13. 4. 5. F 7. C 8. J 9. D Chapter 10 D 14. F 15. A 16. H 17. D 18. H 19. C 20. J B: 4 F D 6. G 7. C J B 18. F 19. B 8. J 9. D 10. H A 13. D 17. 20. 11. J 5. 16. G G G B: 729 11. A20 C Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 6. 10. 4. B 15. 12. D G 14. Page 46 Chapter 10 Assessment Answer Key Form 2C Page 47 Page 48 Sample answer: ∞ ( 2) 1 ∑ 40 - − k=0 2. -28, -40 3. -25x5 11. 2,222,222.2 12. $27,022.87 1 + 5 + 25 + … + 5k - 1 + 5k 13. 4 -7 − 5 5. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 4 1296x4 − 864x3y + 216x2y2 − 24xy3 4 14. + y 89 4. 1 k =− (5 - 1) + 5k 15. 240x3y8 16. 83.7% 17. πi + 2.5416 18. 2e 6 3 6. Answers 1. k 537.3 7. iπ − $38,100 8. ∞ 9. 27 - 9 + 3 - 1 1 1 2 +− -− ; 20 − 3 9 9 (0.031x)n n! 4600∑ − n=0 19. ∞ (3 7 ∑ − x-2 , 3 , 18, 18 √3 6, 6 √ , 54 or 6, -6 √3 18, -18 √ 3 , 54 10. Chapter 10 20. B: A21 n=0 ) n 243 Glencoe Precalculus Chapter 10 Assessment Answer Key Form 2D Page 49 Page 50 Sample answer: ∞ ( 4) 1 ∑ 240 - − 1. k=0 2. -29, -41 3. -49x12 4. 144 11. 42.5 12. $26,723.89 k 7 + 9 + 11 + … + (2k + 5) + (2(k + 1) + 5) = k(k + 6) + 13. (2(k + 1) + 5) 16x4 - 96x3y + 216x2y2 - 216xy3 4 14. + 81y 2 -94 − 5 5. 15. 10x2y16 16. 97.4% 17. πi + 2.5953 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. -4 6. -1092 7. iπ i− √ 2e 4 18. $46,500 8. 9. 81 - 27 + 9 - 3 1 2 +1-− ; 60 − 3 3 ∞ (0.027x)n n! 7300∑ − n=0 19. ∞ (5 ) 3 x-1 ∑ − , 8, -8 √2 , 4, -4 √2 2 , 8, 16 or 4, 4 √ √ 8 2 , 16 10. Chapter 10 20. B: A22 n=0 n 1 Glencoe Precalculus Chapter 10 Assessment Answer Key Form 3 Page 51 Page 52 ⎛ 1 ⎞n ∑ -625 ⎪−⎥ ⎝5⎠ n=2 5 1. 11. about 0.2938 12. 19.8 ft; 220 feet 13. See students’ work. 46 46 − ,− 3 2. 28,125 √ 3 x9y3 376 4. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 14. 5. 23.9 6. 11 5 -230 − 1 4 − x - x3 y + 6x2 y2 16 -16x y3 + 16 y4 15. 5670m12n4 16. 96.8% 17. πi + 3.5473 Answers 3. 9 8 7. 2e 18. 5π i− 6 ⎡ 1500⎢1 + (0.075) + 1 2 1 8 -32 + 8 - 2 + − - −; 9. 10. ⎣ (0.075)2 (0.075)3 ⎤ − + − ; 2 6 ⎦ $1,097,800 8. 5 -25 − ; converge 19. $1616.82 ∞ 8 3 , -12, -4, -4 √ , -36 or -12 √3 -4, 4 √ 3 , -12, 12 √ 3 , -36 Chapter 10 ∑ 20. B: A23 n =0 ( 9x + 22 25 n ) − -5.37824 Glencoe Precalculus Chapter 10 Assessment Answer Key Page 53, Extended-Response Test Sample Answers 1a. Sample answer: Mr. Ling opened a 3b. Here, Sn is defined as a1 - a1rn savings account by depositing $50. He . a1 + a1r + a1r2 + ... + a1r n - 1 = − 1-r plans to deposit $25 more per month into Step 1: Verify that the formula is valid the account. What is his total deposit for n = 1. after three months? The sequence is a1 - a1r1 50 + (n - 1)25, and $100 is his total Since S1 = a1 and S1 = − 1-r deposit after three months. a1(1 - r) = − 1b. Sample answer: The common difference is 1-r $25. The nth term is $50 + (n - 1)$25. = a1 , 1c. Sample answer: the formula is valid for n = 1. 12 S12 = − (50 + 325) = 2250 2 S12 = 50 + 75 + 100 + 125 + 150 + 175 + 200 + 225 + 250 + 275 + 300 + 325 Step 2: Assume that the formula is valid for n = k and derive a formula for n = k + 1. S12 = (50 + 325) + (75 + 300) + (100 + 275) + (125 + 250) + (150 + 225) + (175 + 200) Sk⇒ a1 + a1r + a1r2 + ... + a1r k - 1 a1 - a1rk =− 1-r Since the sums in parentheses are all equal, Sk + 1 ⇒ a1 + a1r + a1r2 + ... a1r k - 1 + a 1r k + 1 - 1 a1 - a1rk =− + a1r k + 1 - 1 12 (50 + 325), or S12 = 6(50 + 325), or − 2 n − (a1+ an). 2 2a. Sample answer: Mimi has $60 to spend on vacation. If she spends half of her money each day, how much will she have left after the third day? (2) 1 $60 × − 3 = −− 1-r a -a r 1 1 = − 1- r a - a r (k + 1) 1-r 1 . 2b. Sample answer: The common ratio is − n-1 The formula gives the same result as adding the (k + 1) term directly. Thus, if the formula is valid for n = k, it is also valid for n = k + 1. Since the formula is valid for n = 2, it is valid for n = 3 and for n = 4, and so on indefinitely. Thus, the formula is valid for all integral values of n. 2 . 2c. Sample answer: 1 11 60 - 60 − 2 S11 = − ≈ 120 1 1-− ( ) 2 1 ; the 2d. If r < 1, the series converges; r = − 2 series converges. 4. From the binomial expansion, the fourth 3a. Prove that the statement is true for n = 1. Then prove that if the statement is true for n, then it is true for n + 1. √ √ 2 A24 6 ( yx yx ) is -y x 6! 20 1 − − −) = -20 · − or - − . ( ) ( x 3!3! y y y -− term of − 2 √ Chapter 10 k+1 1 1 Sk + 1 ⇒ − After the third day, she has $7.50. () 1-r a1 - a1rk + a1r k- a1r k + 1 Apply the original formula for n = k + 1. = $7.50 1 The nth term is 60 − 2 = − + a1r k 3 3 √ 3 3 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 1d. No; arithmetic series have no limits; it is divergent. 1-r a1 - a1rk Chapter 10 Assessment Answer Key Standardized Test Practice Page 54 A B C 9. A B C D 10. F G H J 11. A B C D 12. F G H J 13. A B C D 14. F G H J 15. A B C D D 2. F G H J 3. A B C D 4. F G H J 5. A B C D 6. F G H J 7. A B C D 16. F G H J 8. F G H J 17. A B C D Chapter 10 A25 Answers Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 1. Page 55 Glencoe Precalculus Chapter 10 Assessment Answer Key Standardized Test Practice (continued) Page 56 π ( ) y = ± sin 3x - − + 2 5 18. 10 -1n + 1 3n ∑ − n=1 19. 2n + 1 (-2, 5, 7) 20. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. It is true for n = 1 because 5 = 1(3 + 2). 21. 1 y=− (x - 5)2 - 6 22. 23. 24a. 9 ellipse 0.75 24b. ≈7.127 24c. 1200 Chapter 10 A26 Glencoe Precalculus