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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
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
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
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
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
Lesson Resources
Study Guide and Intervention These
masters provide vocabulary, key concepts,
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
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
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.
Standardized Test Practice These three
pages are cumulative in nature. It includes
two parts: multiple-choice questions with
free-response questions.
• 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
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
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
• 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.
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
• 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
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
• 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
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
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
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
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
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
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
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.
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
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
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
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
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?
week, Drew starts a new reading
10 minutes the first week, 20 minutes
the second, 30 minutes the third,
and so on.
a. How much time will Drew spend
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
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.
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.
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
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
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
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
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
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
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.
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.
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
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
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
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
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
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.
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
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
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
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
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?
d. Repeat parts a through c for at least three more rows of Pascal’s
triangle. What generalization seems to be true?
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.
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
Let x
3!
5!
7!
9!
π
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
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
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
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.
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
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 - −
+−
-−
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
[–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.
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
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
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.
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
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
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?
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
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?
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
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 −
,….
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.
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.
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.
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
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
⎞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
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
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.
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
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
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⎦
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
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.
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
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.
Step 1
10
NAME
0ii_004_PCCRMC10_893811.indd Sec1:3
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.
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
Chapter 10
Series and Sigma Notation
10-1
NAME
-5
417.375
n=0
8. ∑ (2n - 3) 0
n
n=0
n=1
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
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
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
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
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
Enrichment
DATE
1 + an - 1
1 + an - 1
1+1
1
1+−
1+1
1
1+−
1
1+−
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
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
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
x(x + 1) = 1
Factor.
1
x=−
Divide each side by (x + 1).
Solving Equations Using Sequences
10-1
NAME
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
Chapter 10
Arithmetic Sequences
10-2
NAME
Arithmetic Sequences and Series
Study Guide and Intervention
DATE
(continued)
PERIOD
2
2
2
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
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
Glencoe Precalculus
12/4/09 2:31:05 PM
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
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
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
120 min or 2 h
a. How much time will Drew spend
week, Drew starts a new reading
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
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
∑ (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.
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.
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
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
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
(8)
n-1
20.8
∞
16
n=1
Glencoe Precalculus
Glencoe Precalculus
6. ∑ 3n - 1 does not exist
42.227
3/20/09 6:49:27 PM
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
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
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
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?
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
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)
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
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
Glencoe Precalculus
12/4/09 2:48:55 PM
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.
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
DATE
PERIOD
(k + 1) is a common factor.
Add 2(k + 1) to both sides.
Replace n with k.
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
Chapter 10
Extended Mathematical Induction
10-4
NAME
Mathematical Induction
Practice
DATE
PERIOD
3
3
3
6
3
3
3
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
Chapter 10
3
A11
24
Glencoe Precalculus
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
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
Enrichment
DATE
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
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.
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
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
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
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
Glencoe Precalculus
12/5/09 3:25:04 PM
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
1.6%
9.4%
23.4%
31.3%
23.4%
9.4%
1.6%
0
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
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.
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
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
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
Glencoe Precalculus
12/4/09 3:20:13 PM
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 = −
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
∞
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
005_036_PCCRMC10_893811.indd 33
Chapter 10
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
(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
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
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
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
2. Create an alternating series, other than a geometric series with negative
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
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
Glencoe Precalculus
12/4/09 3:24:37 PM
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 - −
+−
-−
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
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
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.
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
Page 37
Form 1
Page 41
1.
2.
3.
1.
2.
geometric sequence
4.
explicit sequence
5.
3.
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
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
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
6.
10.
4.
B
15.
12.
D
G
14.
Page 46
Form 2C
Page 47
Page 48
∞
( 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.
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.
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
Form 2D
Page 49
Page 50
∞
( 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
-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
Form 3
Page 51
Page 52
⎛ 1 ⎞n
∑ -625 ⎪−⎥
⎝5⎠
n=2
5
1.
11.
12.
19.8 ft; 220 feet
13.
See students’
work.
46 46
−
,−
3
2.
28,125 √
3 x9y3
376
4.
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
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
Page 53, Extended-Response Test
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 ,
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
.
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
1d. No; arithmetic series have no limits; it is
divergent.
1-r
a1 - a1rk
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
1.
Page 55
Glencoe Precalculus
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.
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
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