7-8 Recursive Formulas. Find the first five terms of each sequence. 1. SOLUTION: Use a 1. = 16 and the recursive formula to find the next four terms.
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1 Find the first five terms of each sequence. 1. Use a 1 = 16 and the recursive formula to find the next four terms. The first five terms are 16, 13, 10, 7, and 4. esolutions Manual - Powered by Cognero Page 1
2 2. Use a 1 = 5 and the recursive formula to find the next four terms. The first five terms are 5, 10, 30, 110, and 430. Write a recursive formula for each sequence. 3. 1, 6, 11, 16, = 5; 11 6 = 5, = 5 There is a common difference of 5. The sequence is arithmetic. Use the formula for an arithmetic sequence. The first term a 1 is 1, and n 2. A recursive formula for the sequence 1, 6, 11, 16, is a 1 = 1, a n = a n 1 + 5, n 2. esolutions Manual - Powered by Cognero Page 2
3 4. 4, 12, 36, 108, = 8; = 24, = 72 There is no common difference. Check for a common ratio by dividing each term by the term that precedes it. = 3 ; = 3 ; = 3 There is a common ratio of 3. The sequence is geometric. Use the formula for a geometric sequence. The first term a 1 is 4, and n 2. A recursive formula for the sequence 4, 12, 36, 108, is a 1 = 4, a n = 3a n 1, n 2. esolutions Manual - Powered by Cognero Page 3
4 5. BALL A ball is dropped from an initial height of 10 feet. The maximum heights the ball reaches on the first three bounces are shown. a. Write a recursive formula for the sequence. b. Write an explicit formula for the sequence. a. The sequence of heights is 10, 6, 3.6, and = 4; = -2.4, = There is no common difference. Check for a common ratio by dividing each term by the term that precedes it. ; ; There is a common ratio of 0.6. The sequence is geometric. Use the formula for a geometric sequence. The first term a 1 is 10, and n 2. A recursive formula for the sequence 10, 6, 3.6, and 2.16, is a 1 = 10, a n = 0.6a n 1, n 2. b. Use the formula for the nth terms of a geometric sequence. The explicit formula is a n = 10(0.6) n 1. esolutions Manual - Powered by Cognero Page 4
5 For each recursive formula, write an explicit formula. For each explicit formula, write a recursive formula. 6. The common difference is 16. Use the formula for the nth terms of an arithmetic sequence. The explicit formula is a n = 16n a n = 5n + 8 Write out the first 4 terms. 13, 18, 23, = 5; = 5, = 5 There is a common difference of 5. The sequence is arithmetic. Use the formula for an arithmetic sequence. The first term a 1 is 13, and n 2. A recursive formula for the explicit formula a n = 5n + 8 is a 1 = 13, a n = a n 1 + 5, n a n = 15(2) n 1 Write out the first 4 terms. 15, 30, 60, = 30; = 30, = 60 There is no common difference. Check for a common ratio by dividing each term by the term that precedes it. There is a common ratio of 2. The sequence is geometric. Use the formula for a geometric sequence. The first term a 1 is 15, and n 2. A recursive formula for the explicit formula a n = 15(2) n - 1 is a1 = 15, a n = 2a n 1, n 2. esolutions Manual - Powered by Cognero Page 5
6 9. The common ratio is 4. Use the formula for the nth terms of a geometric sequence. The explicit formula is a n = 22(4) n 1. Find the first five terms of each sequence. 10. Use a 1 = 23 and the recursive formula to find the next four terms. The first five terms are 23, 30, 37, 44, and 51. esolutions Manual - Powered by Cognero Page 6
7 11. Use a 1 = 48 and the recursive formula to find the next four terms. The first five terms are 48, 16, 16, 0, and Use a 1 = 8 and the recursive formula to find the next four terms. The first five terms are 8, 20, 50, 125, and esolutions Manual - Powered by Cognero Page 7
8 13. Use a 1 = 12 and the recursive formula to find the next four terms. The first five terms are 12, 15, 24, 51, and 132. esolutions Manual - Powered by Cognero Page 8
9 14. Use a 1 = 13 and the recursive formula to find the next four terms. The first five terms are 13, 29, 55, 113, and 223. esolutions Manual - Powered by Cognero Page 9
10 15. Use and the recursive formula to find the next four terms. The first five terms are. Write a recursive formula for each sequence = 13; 14 ( 1) = 13, 27 ( 14) = 13 There is a common difference of 13. The sequence is arithmetic. Use the formula for an arithmetic sequence. The first term a 1 is 12, and n 2. A recursive formula for the sequence 12, 1, 14, 27, is a 1 = 12, a n = a n 1 13, n 2. esolutions Manual - Powered by Cognero Page 10
11 17. 27, 41, 55, 69, = 14; = 14, = 14 There is a common difference of 14. The sequence is arithmetic. Use the formula for an arithmetic sequence. The first term a 1 is 27, and n 2. A recursive formula for the sequence 27, 41, 55, 69, is a 1 = 27, a n = a n , n , 11, 20, 29, = 9; = 9, = 9 There is a common difference of 9. The sequence is arithmetic. Use the formula for an arithmetic sequence. The first term a 1 is 2, and n 2. A recursive formula for the sequence 2, 11, 20, 29, is a 1 = 2, a n = a n 1 + 9, n , 80, 64, 51.2, = 20; = 16, = 12.8 There is no common difference. Check for a common ratio by dividing each term by the term that precedes it. There is a common ratio of 0.8. The sequence is geometric. Use the formula for a geometric sequence. The first term a 1 is 100, and n 2. A recursive formula for the sequence 100, 80, 64, 51.2, is a 1 = 100, a n = 0.8a n 1, n 2. esolutions Manual - Powered by Cognero Page 11
12 = 100; 90 ( 60) = 30, = 225 There is no common difference. Check for a common ratio by dividing each term by the term that precedes it. There is a common ratio of 1.5. The sequence is geometric. Use the formula for a geometric sequence. The first term a 1 is 40, and n 2. A recursive formula for the sequence 40, 60, 90, 135, is a 1 = 40, a n = 1.5a n 1, n , 27, 9, 3, = 54; 9 27 = 18, 3 9 = 6 There is no common difference. Check for a common ratio by dividing each term by the term that precedes it. There is a common ratio of. The sequence is geometric. Use the formula for a geometric sequence. The first term a 1 is 81, and n 2. A recursive formula for the sequence 81, 27, 9, 3, is a 1 = 81, a n = a n 1, n 2. esolutions Manual - Powered by Cognero Page 12
13 22. CCSS MODELING A landscaper is building a brick patio. Part of the patio includes a pattern constructed from triangles. The first four rows of the pattern are shown. a. Write a recursive formula for the sequence. b. Write an explicit formula for the sequence. 23. For each recursive formula, write an explicit formula. For each explicit formula, write a recursive formula. Write out the first 4 terms: 3, 12, 48, = 9; = 36, = 144 There is no common difference. Check for a common ratio by dividing each term by the term that precedes it. There is a common ratio of 4. The sequence is geometric. Use the formula for a geometric sequence. The first term a 1 is 3, and n 2. A recursive formula for the explicit formula a n = 15(2) n 1 is a 1 = 3, a n = 4a n 1, n The common difference is 12. Use the formula for the nth terms of an arithmetic sequence. The explicit formula is a n = 12n esolutions Manual - Powered by Cognero Page 13
14 25. The common ratio is. Use the formula for the nth terms of a geometric sequence. The explicit formula is. 26. Write out the first 4 terms. 45, 38, 31, = 7; = 7, = 7 There is a common difference of 7. The sequence is arithmetic. Use the formula for an arithmetic sequence. The first term a 1 is 45, and n 2. A recursive formula for the explicit formula a n = 7n + 52 is a 1 = 45, a n = a n 1 7, n 2. esolutions Manual - Powered by Cognero Page 14
15 27. TEXTING Barbara received a chain text that she forwarded to five of her friends. Each of her friends forwarded the text to five more friends, and so on. a. Find the first five terms of the sequence representing the number of people who receive the text in the nth round. b. Write a recursive formula for the sequence. c. If Barbara represents a 1, find a 8. a. Then the first 5 terms of the sequence would be 1, 5, 25, 125, 625. b. The first term a 1 is 1, and n 2. A recursive formula is a 1 = 1, a n = 5a n 1, n 2. c. On the 8th round, 78,125 would receive the chain text message. esolutions Manual - Powered by Cognero Page 15
16 28. GEOMETRY Consider the pattern below. The number of blue boxes increases according to a specific pattern. a. Write a recursive formula for the sequence of the number of blue boxes in each figure. b. If the first box represents a 1, find the number of blue boxes in a 8. a. The sequence of blue boxes is 0, 4, 8, and = 4; 8 4 = 4, 12 8 = 4 There is a common difference of 4. The sequence is arithmetic. Use the formula for an arithmetic sequence. The first term a 1 is 0, and n 2. A recursive formula is a 1 = 0, a n = a n 1 + 4, n 2. b. Use the formula for the nth terms of an arithmetic sequence. When n = 8, there will be 28 blue boxes. esolutions Manual - Powered by Cognero Page 16
17 29. TREE The growth of a certain type of tree slows as the tree continues to age. The heights of the tree over the past four years are shown. a. Write a recursive formula for the height of the tree. b. If the pattern continues, how tall will the tree be in two more years? Round your answer to the nearest tenth of a foot. a. The sequence of heights is 10, 11, 12.1, and = 11; = 1.1, = 1.21 There is no common difference. Check for a common ratio by dividing each term by the term that precedes it. There is a common ratio of 1.1. The sequence is geometric. Use the formula for a geometric sequence. The first term a 1 is 10, and n 2. A recursive formula for the sequence 10, 11, 12.1, 13.31, is a 1 = 10, a n = 1.1a n 1, n 2. b. Use the formula for the nth terms of a geometric sequence. In two more years, the tree will be 16.1 feet tall. esolutions Manual - Powered by Cognero Page 17
18 30. MULTIPLE REPRESENTATIONS The Fibonacci sequence is neither arithmetic nor geometric and can be defined by a recursive formula. The first terms are 1, 1, 2, 3, 5, 8, a. Logical Determine the relationship between the terms of the sequence. What are the next five terms in the sequence? b. Algebraic Write a formula for the nth term if a 1 = 1, a 2 = 1, and n 3. c. Algebraic Find the 15th term. d. Analytical Explain why the Fibonacci sequence is not an arithmetic sequence. a. Sample answer: The first two terms are 1. Starting with the third term, the two previous terms are added together to get the next term. So, the next 5 terms after 8 is or 13, or 21, or 34, or 55, and or 89. b. The first term a 1 is 1 and the second term a 2 is 1, and n 3. A recursive formula for Fibonacci sequence is a 1 = 1, a 2 = 1, a n = a n ÿ 2 + a n ÿ 1, n 3. c. First, find the 13th and 14th terms by writing out 14 terms of the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377. Then, a 13 = 233 and a 14 = 377. Thus, a 15 = a 13 + a 14 or 610. d. Sample answer: Fibonacci sequence is not an arithmetic sequence since there is no common difference. 31. ERROR ANALYSIS Patrick and Lynda are working on a math problem that involves the sequence 2, 2, 2, 2, 2,. Patrick thinks that the sequence can be written as a recursive formula. Lynda believes that the sequence can be written as an explicit formula. Is either of them correct? Explain. Both; sample answer: The sequence can be written as the recursive formula a 1 = 2, a n = ( 1)a n 1, n 2. The sequence can also be written as the explicit formula a n = 2( 1) n 1. esolutions Manual - Powered by Cognero Page 18
19 32. CHALLENGE Find a 1 for the sequence in which a 4 = 1104 and a n = 4a n Find a 3 first. Find a 2 next. Find a 1. Therefore, a 1 is CCSS ARGUMENTS Determine whether the following statement is true or false. Justify your reasoning. There is only one recursive formula for every sequence. False; sample answer: A recursive formula for the sequence 1, 2, 3, can be written as a 1 = 1, a n = a n , n 2 or as a 1 = 1, a 2 = 2, a n = a n , n 3. esolutions Manual - Powered by Cognero Page 19
20 34. CHALLENGE Find a recursive formula for 4, 9, 19, 39, 79, 9 4 = 5; 19 9 = 10, = 20 There is no common difference. Check for a common ratio by dividing each term by the term that precedes it. There is no common ratio. Therefore the sequence must be a combination of both. From the difference above, you can see each is twice as big as the previous. So r is 2. From the ratios, if each numerator was one less, the ratios would be 2. Thus, the common difference is 1. So, if the first term a 1 is 4, and n 2 a recursive formula for the sequence 4, 9, 19, 39, 79, is a 1 = 4, a n = 2a n 1 + 1, n WRITING IN MATH Explain the difference between an explicit formula and a recursive formula. Sample answer: In an explicit formula, the nth term a n is given as a function of n. In a recursive formula, the nth term a n is found by performing operations to one or more of the terms that precede it. 36. Find a recursive formula for the sequence 12, 24, 36, 48,. A B C D = 12; = 12, = 12 There is a common difference of 12. The sequence is arithmetic. Use the formula for an arithmetic sequence. The first term a 1 is 12, and n 2. A recursive formula for the sequence 12, 24, 36, 48, is a 1 = 12, a n = a n , n 2. Therefore, the correct choice is C. esolutions Manual - Powered by Cognero Page 20
21 37. GEOMETRY The area of a rectangle is 36 m 4 n 6 square feet. The length of the rectangle is 6 m 3 n 3 feet. What is the width of the rectangle? F 216m 7 n 9 ft G 6mn 3 ft H 42m 7 n 3 ft J 30mn 3 ft Use the area formula for a rectangle to determine the measure of the width. Therefore, the correct choice is G. 38. Find an inequality for the graph shown. A B C D The y-intercept of the line is 4. The slope is 2. The graph is shaded below, so the inequality should use a < or symbol. Since the line is dashed, the inequality does not include the equals. Therefore, the correct choice is C. esolutions Manual - Powered by Cognero Page 21
22 39. Write an equation of the line that passes through ( 2, 20) and (4, 58). F y = 13x + 6 G y = 19x 18 H y = 19x + 18 J y = 13x 6 First find the slope. Use the point-slope formula to find the equation. Therefore, the correct choice is F. Find the next three terms in each geometric sequence , 225, 75,... Find the common ratio. The common ratio is. Multiply by the common ratio to find next three terms. They are , 24, 36,... Find the common ratio. The common ratio is. Multiply by the common ratio to find next three terms. They are -54, 81, esolutions Manual - Powered by Cognero Page 22
23 42. 6, 18, 54,... Find the common ratio. The common ratio is 3. Use the common ratio to find next three terms. They are 162, 324, , 256, 128,... Find the common ratio. The common ratio is. Use the common ratio to find next three terms. They are -64, 32, , 25, 5,... Find the common ratio. The common ratio is. Use the common ratio to find next three terms. They are , 60, 300,... Find the common ratio. The common ratio is 5. Multiply by the common ratio to find next three terms. They are 1500, 7500, 37, INVESTMENT Nicholas invested $2000 with a 5.75% interest rate compounded monthly. How much money will Nicholas have after 5 years? After 5 years, Nicholas will have $ esolutions Manual - Powered by Cognero Page 23
24 47. TOURS The Snider family and the Rollins family are traveling together on a trip to visit a candy factory. The number of people in each family and the total cost are shown in the table below. Find the adult and children s admission prices. Let a be the admissions cost for adults and c the admission cost of children. Then 2a + 3c = 58 and 2a + c = 38. So, tickets for adults cost $14 and tickets for children cost $10. Write each equation in standard form esolutions Manual - Powered by Cognero Page 24
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26 54. Simplify each expression. If not possible, write simplified There are no like terms, so the expression is simplified. 59. There are no like terms, so the expression is simplified. esolutions Manual - Powered by Cognero Page 26
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