Name Class Date. Understanding Sequences

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1 Name Class Date 5-1 Introduction to Sequences Going Deeper Essential question: Why is a sequence a function? 1 MCC9 1.F.IF. ENGAGE Understanding Sequences Video Tutor A sequence is an ordered list of numbers or other items. Each element in a sequence is called a term. For instance, in the sequence 1,, 5, 7, 9,..., the second term is. Each term in a sequence can be paired with a position number, and these pairings establish a function whose domain is the set of position numbers and whose range is the set of terms, as illustrated below. The position numbers are consecutive integers that typically start at either 1 or 0. Position number n Domain Term of sequence f(n) Range For the sequence shown in the table, you can write f (4) = 7, which can be interpreted as the fourth term of the sequence is 7. REFLECT 1a. The domain of the function f defining the sequence, 5, 8, 11, 14,... is the set of consecutive integers starting with 0. What is f (4)? Explain how you determined your answer. 1b. How does your answer to Question 1a change if the domain of the function is the set of consecutive integers starting with 1? 1c. Predict the next term in the sequence 48, 4, 6, 0, 4,.... Explain your reasoning. 1d. Why is the relationship between the position numbers and the terms of a sequence a function? 1e. Give an example of a sequence from your everyday life. Explain why your example represents a sequence. Module 5 15 Lesson 1

2 Some numerical sequences can be described by using algebraic rules. An explicit rule for a sequence defines the nth term as a function of n. MCC9 1.F.IF. EXAMPLE Using an Explicit Rule to Generate a Sequence Write the first 4 terms of the sequence f(n) = n + 1. Assume that the domain of the function is the set of consecutive integers starting with 1. n n + 1 f(n) = = = = + 1 The first 4 terms are. REFLECT a. How could you use a graphing calculator to check your answer? b. Explain how to find the 0th term of the sequence. A recursive rule for a sequence defines the nth term by relating it to one or more previous terms. MCC9 1.F.IF. EXAMPLE Using a Recursive Rule to Generate a Sequence Write the first 4 terms of the sequence with f(1) = and f(n) = f(n - 1) + for n. Assume that the domain of the function is the set of consecutive integers starting with 1. The first term is given: f (1) =. Use f (1) to find f (), f () to find f (), and so on. In general, f (n - 1) refers to the term that precedes f (n). n f(n - 1) + f(n) f( - 1) + = f(1) + = + f ( - 1 ) + = f ( ) + = + 4 f ( - 1 ) + = f ( ) + = + The first 4 terms are. Module 5 16 Lesson 1

3 REFLECT a. Describe how to find the 1th term of the sequence. b. Suppose you want to find the 50th term of a sequence. Would you rather use a recursive rule or an explicit rule? Explain your reasoning. 4 MCC9 1.F.BF.1a EXAMPLE Modeling a Sequence A male honeybee has one female parent, and a female honeybee has one male and one female parent. In the diagram below, a male honeybee is represented by M in row 1. His parent is represented by F in row. Her parents are represented by M and F in row, and so on. Write a recursive rule for a sequence that describes the number of bees in each row. A Extend the diagram to show rows 5, 6, and 7. Row 1 M Row F Row M F Row 4 F M F Row 5 Row 6 Row 7 B Complete the table to show the number of bees in each row. Row (position number) Number of bees (term of sequence) 1 1 Module 5 17 Lesson 1

4 C Write a recursive rule for the sequence in the table. Assume that the domain of the function is the set of consecutive integers starting with 1. First, write the rule in words. The first two terms are both. Every other term is the of the previous two terms. Then, write the rule algebraically. f (1) = f () = and The first and second terms are both 1. f (n) = f ( n - ) + f (n - ) for n Each successive term is the sum of the preceding two terms. REFLECT 4a. If you continued the pattern in the diagram, how many bees would be in the 8th row? Explain how you determined your answer. 4b. The sequence given in the table, 1, 1,,, 5, 8, 1,..., is called the Fibonacci sequence. An explicit rule for the Fibonacci sequence is f (n) = 1 Ç 5 ( 1 + Ç n 5 ) - 1 Ç 5 ( 1 - Ç n 5 ) where the values of n are consecutive integers starting with 1. Use the explicit rule to show that f (1) = 1. Then use a calculator and the explicit rule to find the 9th term of the Fibonacci sequence. 4c. Now use the recursive rule to find the 9th term of the Fibonacci sequence. Does your result agree with the result from the explicit rule? 4d. Which rule for the Fibonacci sequence would be easier to use if you did not have a calculator? Explain. 4e. The number of petals on many flowers is equal to a Fibonacci number, that is, one of the terms in the Fibonacci sequence. Based on this fact, is a flower more likely to have 0 petals or 1 petals? Explain. Module 5 18 Lesson 1

5 practice Write the first four terms of each sequence. Assume that the domain of the function is the set of consecutive integers starting with f (n) = (n - 1). f (n) = n + 1 n +. f (n) = 4(0. 5) n 4. f (n) = ÇÇÇ n f (1) = and f (n) = f (n - 1) + 10 for n 6. f (1) = 16 and f (n) = 1 f (n - 1) for n 7. f (1) = 1 and f (n) = f (n - 1) + 1 for n 8. f (1) = f () = 1 and f (n) = f (n - ) - f (n - 1) for n 9. Each year for the past 4 years, Donna has gotten a raise equal to 5% of the previous year s salary. Her starting salary was $40,000. a. Complete the table to show Donna s salary over time. b. Write a recursive rule for the sequence in the table. Assume that the domain of the function is the set of consecutive integers starting with 0, so the first term of the sequence is f (0). Year (position number) Salary ($) (term of sequence) 0 40,000 1 c. What is f (7), rounded to the nearest whole number? What does f (7) represent in this situation? Write the 1th term of each sequence. Assume that the domain of the function is the set of consecutive integers starting with f (n) = n f (n) = n(n + 1) Module 5 19 Lesson 1

6 1. The diagram shows the first four figures in a pattern of dots. a. Draw the next figure in the pattern. b. Use the pattern to complete the table. c. Write an explicit rule for the sequence in the table. Assume that the domain of the function is the set of consecutive integers starting with 1. d. How many dots will be in the 10th figure of the pattern? Figure (position number) Number of dots (term of sequence) Write an explicit rule for each sequence. Assume that the domain of the function is the set of consecutive integers starting with n f(n) n f(n) n f(n) n f(n) n f(n) Write a recursive rule for each sequence. Assume that the domain of the function is the set of consecutive integers starting with n f(n) Module Lesson 1

7 Name Class Date Additional Practice 5-1 Find the first 5 terms of each sequence. 1. a 1 = 1, a n = (a n 1 ). a 1 =, a n = (a n 1 + 1) 5. a 1 =, a n = (a n 1 ) 1 4. a 1 = 1, a n = 6 (a n 1 ) 5. a 1 = 1, a n = (a n 1 1) an 1 6. a1 =, an = 7. a n = (n )(n + 1) 8. a n = n(n 1) 9. a n = n n 10. a n 1 = n 11. a n = ( ) n 1 1. a n = n n Write a possible explicit rule for the nth term of each sequence. 1. 8, 16, 4,, 40, , 0.4, 0.9, 1.6,.5, 15., 6, 11, 18, 7, 16.,,,,, , 1, 4, 7, 10, 18. 5, 1, 0., 0.04, 0.008, Solve. 19. Find the number of line segments in the next two iterations. 0. Jim charges $50 per week for lawn mowing and weeding services. He plans to increase his prices by 4% each year. a. Graph the sequence. b. Describe the pattern. c. To the nearest dollar, how much will he charge per week in 5 years? Module Lesson 1

8 Problem Solving Tina is working on some home improvement projects involving repeated tasks. She wants to analyze her work patterns. 1. Tina is hammering nails into wallboard. With the first hit, a nail goes in 1.5 millimeters; with the second, it goes in an additional 9 mm; with the third, it goes in an additional 6 mm; and with the fourth it goes in 4 mm further. Suppose this pattern continues. Predict how far the nail would go in with the seventh hit. a. Complete the table to find first differences, second differences, and ratios. Distance 1.5mm 9mm 6mm 4mm Ratios First Differences 4.5 Second Differences b. How do you know whether the rule for the sequence of distances that the nail goes in is linear, quadratic, or exponential? c. Write a possible rule for a n, the nth term in the sequence. d. If this pattern continues, how far would the nail go in with the seventh hit?. Tina builds a fence for her neighbor. It takes her 10 minutes to pound the first fence post into the ground. The neighbor predicts that Tina should improve her time on each successive fence post according to the rule a n = F (n 1), where F is the time for the first fence post, and a n is the time it takes to pound in the nth post. a. Use the rule to find the time it should take Tina to pound each of the first 4 fence posts into the ground. b. If the rule that describes Tina s time on each successive post is a n = F 1. 5 n 1, how long will it take her to pound the fourth fence post into the ground? The label on Pete s blue jeans states that, when washed, the jeans will lose 5% of their color. Choose the letter for the best answer.. Which rule describes the percent of color left in the blue jeans after n washings? A a n = 100 (0.05) n B a n = 100 (0.95) n C a n = 100 (0.95) n D a n = 100 (0.05) n 4. How much of the original color will be left after 8 washings? F 66% G 60% H 40% J 4% Module 5 14 Lesson 1