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7-4 Exponential Models in Recursive Form TEKS FOCUS VOCABULARY TEKS (5)(B) Formulate exponential and logarithmic equations that model realworld situations, including exponential relationships written in recursive notation. ĚExplicit formula – An explicit TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate. ĚRecursive formula – A recursive formula describes the nth term of a sequence using the number n. formula relates each term of a sequence after the first term to the term before it. ĚSequence – A sequence is an ordered list of numbers. Additional TEKS (1)(A) ĚTerm of a sequence – Each number in a sequence is a term of the sequence. ĚImplication – a conclusion that follows from previously stated ideas or reasoning without being explicitly stated ĚRepresentation – a way to display or describe information. You can use a representation to present mathematical ideas and data. ESSENTIAL UNDERSTANDING If the numbers in a list follow a pattern, you may be able to use a rule to relate each number in the list to its numerical position. Key Concept Exponentiation in Recursive Form Sometimes you can see the pattern in a sequence by comparing each term to the one that came before it. For example, in the sequence 133, 130, 127, 124, . . . , each term after the first term is equal to three less than the previous term. A recursive definition for this sequence contains two parts. (a) an initial condition (the value of the first term): a1 = 133 (b) a recursive formula (relates each term after the first term to the one before it): an = an-1 - 3, for n 7 1 You can use properties of exponents to write patterns based on exponentiation in recursive form. a1 = x 1 = x an = xn = x x(n-1) = x an-1 # 286 Lesson 7-4 # Exponential Models in Recursive Form Problem 1 P TEKS Process Standard (1)(D) Formulating Recursive Exponential Functions The table shows the number of subscribers to an artist’s photo-sharing account over a period of several months. Write an explicit formula and a recursive formula to model the data. Month Subscribers 1 2 3 4 5 7 21 63 189 567 Step 1 Write an explicit formula. Write each term of the sequence with an exponent and look for a pattern. a1 = 7 =7 How can you check the formula? Check that the formula works for specific values of n, including n = 1. For n = 1, a1 = 7(3)1-1 = 7(3)0 = 7. # a2 = 21 30 =7 # a3 = 63 31 =7 # a4 = 189 32 =7 # a5 = 567 33 =7 # 34 In each term, the exponent is 1 less than the term number. The explicit formula is an = 7(3)n-1 . Step 2 Write a recursive formula. S Write the first few terms of the sequence using the previous term and look for a pattern. a1 = 7 # 7 = 3a1 a3 = 63 = 3 # 21 = 3a2 a4 = 189 = 3 # 63 = 3a3 a2 = 21 = 3 The recursive formula is a1 = 7 and an = 3an-1 . PearsonTEXAS.com 287 Problem 2 P Writing Exponential Functions in Recursive Form The function y = 8 ~ 2x models the number of E. coli cells y in a petri dish x hours after the start of an experiment. Write a recursive formula to model the situation. Let an represent the number of cells after n hours. Write the first few terms of the sequence using the previous term and look for a pattern. What does the recursive formula tell you about the situation? The formula says that there are 16 cells after the first hour and the number of cells doubles every hour after that. # 21 = 16 a2 = 8 # 22 = 32 = 2 # 16 = 2a1 a3 = 8 # 23 = 64 = 2 # 32 = 2a2 a4 = 8 # 24 = 128 = 2 # 64 = 2a3 a1 = 8 The recursive formula is a1 = 16 and an = 2an-1 . T Problem P bl 3 TEKS Process Standard (1)(A) Using a Recursive Exponential Function to Model a Situation You invest $500 in a savings account that pays 2% annual interest. Write an explicit formula and a recursive formula to model the situation. How do you find the amount in the account after 1 year? After 1 year, you have the starting amount plus 2% of the starting amount, or 500 + 500(0.02) = 500(1.02). Step 1 Define the variables and write the first few terms of the sequence. S Let n represent the number of years. Let an represent the amount in the account after n years. # 1.02 a2 = 500 # 1.02 # 1.02 a3 = 500 # 1.02 # 1.02 # 1.02 a1 = 500 Step 2 Write an explicit formula. In the above expressions for an , the factor 1.02 appears n times. an = 500(1.02)n Step 3 Write a recursive formula. In the above expressions for an , a1 = 500 1.02 times the previous term. a1 = 510 and an = 1.02an-1 288 Lesson 7-4 Exponential Models in Recursive Form # 1.02 = 510 and each term is HO ME RK O NLINE WO PRACTICE and APPLICATION EXERCISES For additional support when completing your homework, go to PearsonTEXAS.com. Scan page for a Virtual Nerd™ tutorial video. 1. The table shows the number of visitors to a Web page over a period of several months. Write an explicit formula and a recursive formula to model the data. Month Visitors 2. The table shows the profit made by a small catering company over a period of several years. Write an explicit formula and a recursive formula to model the data. 1 2 3 4 5 17 34 68 136 272 2 3 4 5 1 Year Profit ($) 5200 3. A student repeatedly folds a sheet of paper in half. The table shows the area of each of the congruent regions formed by the creases. Write an explicit formula and a recursive formula to model the data. Number of Folds 1 Area of Each 256 Region (cm2) 7800 11,700 17,550 26,325 1 fold 2 congruent regions 2 3 4 5 128 64 32 16 2 folds 4 congruent regions # 4. The function y = 25 2x models the jackpot y, in dollars, on a game show after the show has been on the air for x weeks. Write a recursive formula to model the situation. 5. The function y = 3500(1.1)x models the value y, in dollars, of a piece of artwork after x years. Write a recursive formula to model the situation. 6. The function y = 18,000(0.85)x models the value y, in dollars, of a car after x years. Write a recursive formula to model the situation. 7. You invest $400 in a savings account that pays 2.5% annual interest. Write an explicit formula and a recursive formula to model the situation. 8. An employee joins a company at the start of the year and earns a salary of $40,000. At the end of each year, the employee receives a 4% raise. Write an explicit formula and a recursive formula to model the situation. 9. An accountant buys a new computer for $1200. Each year, the value of the computer decreases by 20%. Write an explicit formula and a recursive formula to model the situation. 10. Explain Mathematical Ideas (1)(G) The value of a collectible baseball card is currently $620 and its value is expected to increase by 5% each year. A student modeled the situation by writing the recursive formula a1 = 620 and an = (1.05)an-1 , where an represents the value of the baseball card after n years. Is the student’s formula correct? Explain. PearsonTEXAS.com 289 11. The fractal known as the Sierpinski carpet begins with a square. At each subsequent stage, every square is divided into nine congruent squares and the center square is removed. Assume the area of the square in Stage 1 is 1 square unit. Write an explicit formula and a recursive formula to model the area an of the figure in the nth stage of the fractal. Stage 1 Stage 2 12. The owner of a corner store finds that a 1 Week juice drink suddenly becomes popular with Revenue From students at a school across the street. The 4 Juice Drinks ($) store’s owner records the revenue from the drinks over a period of several weeks, but she does not do so every week. The table shows the data. Write an explicit formula and a recursive formula to model the data. 3 4 6 36 108 972 # 13. Analyze Mathematical Relationships (1)(F) A real-world situation is modeled by the explicit formula an = p qn-1 for real numbers p and q with p ≠ 0, q ≠ 0, and q ≠ 1. Write a recursive formula to model the situation. TEXAS Test Practice T 14. The amount of money in a bank account is modeled by the explicit formula an = 320(1.035)n-1 , where an is the amount of money in the account, in dollars, at the end of n years. Which is a true statement about the situation? A. The initial investment in the account is $331.20. B. The account pays 35% annual interest. C. In a recursive formula for the situation, a1 = 331.2. D. The amount of money in the account at the end of year 5 is $367.21. 15. Which of the following is a recursive formula for an exponential relationship? F. a1 = 2 and an = 4 + an-1 G. a1 = 4 and an = 0.5an-1 H. a1 = 0.5 and an = n an-1 J. a1 = 4 and an = (an-1)2 16. The number of employees at a small software company grows according to an exponential relationship. After 2 years, the company has 18 employees. After 4 years, the company has 72 employees. If an represents the number of employees after n years, which recursive formula models the situation? A. a1 = 18 and an = 4an-1 C. a1 = 9 and an = 2an-1 B. a1 = 18 and an = 2an-1 D. a1 = 9 and an = 4an-1 17. Describe a real-world situation that can be modeled by the recursive formula a1 = 6 and an = 3an-1 . 290 Lesson 7-4 Exponential Models in Recursive Form Stage 3