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P. 692 Concept

an + 1 = 2an + 7 Recursive formula a1 + 1 = 2a1 + 7 n = 1 Use a Recursive Formula Find the first five terms of the sequence in which a1 = 5 and an + 1 = 2an + 7, n ≥ 1. an + 1 = 2an + 7 Recursive formula a1 + 1 = 2a1 + 7 n = 1 a2 = 2(5) + 7 or 17 a1 = 5 a2 + 1 = 2a2 + 7 n = 2 a3 = 2(17) + 7 or 41 a2 = 17 a3 + 1 = 2a3 + 7 n = 3 a4 = 2(41) + 7 or 89 a3 = 41 Example 1

Answer: The first five terms of the sequence are 5, 17, 41, 89, 185. Use a Recursive Formula a4 + 1 = 2a4 + 7 n = 4 a5 = 2(89) + 7 or 185 a4 = 89 Answer: The first five terms of the sequence are 5, 17, 41, 89, 185. Example 1

Find the first five terms of the sequence in which a1 = 2 and an + 1 = 3an + 2, n ≥ 1. B. 2, 8, 26, 80, 242 C. 2, 6, 20, 62, 188 D. 26, 80, 242, 728, 2186 Example 1

A. Write a recursive formula for the sequence. 3, 10, 17, 24, 31, … Write Recursive Formulas A. Write a recursive formula for the sequence. 3, 10, 17, 24, 31, … Step 1 Determine whether the sequence is arithmetic or geometric. The sequence is arithmetic because each term after the first can be found by adding the common difference. Step 2 Find the common difference. d = 10 – 3 or 7 Example 2A

Step 3 Write a recursive formula. Write Recursive Formulas Step 3 Write a recursive formula. an = an–1 + d Recursive formula for arithmetic sequence an = an–1 + 7 d = 7 Answer: A recursive formula for the sequence is an = an–1 + 7, where a1 = 3. Example 2A

B. Write a recursive formula for the sequence. 5, 20, 80, 320, 1280, … Write Recursive Formulas B. Write a recursive formula for the sequence. 5, 20, 80, 320, 1280, … Step 1 Determine whether the sequence is arithmetic or geometric. The sequence is geometric because each term after the first can be found after multiplying by the common ratio. Step 2 Find the common ratio. Example 2B

Step 3 Write a recursive formula. Write Recursive Formulas Step 3 Write a recursive formula. an = r ● an–1 Recursive formula for geometric sequence an = 4 ● an–1 r = 4 Answer: A recursive formula for the sequence is an = 4 ● an–1, where a1 = 5. Example 2B

C. Write a recursive formula for the sequence. a3 = 6 and d = 5 Write Recursive Formulas C. Write a recursive formula for the sequence. a3 = 6 and d = 5 Step 1 Determine whether the sequence is arithmetic or geometric. Because d is given, the sequence is arithmetic. Step 2 Write a recursive formula. an = an–1 + d Recursive formula for arithmetic sequence an = an–1 + 5 d = 5 Answer: A recursive formula for the sequence is an = an–1 + 5, where a1 = –4. Example 2C

A. Write a recursive formula for the sequence. 6, 17, 28, 39, 50, … A. a1 = 6; an = an–1 + 11 B. a1 = 6; an = 6 ● an–1 + 11 C. a1 = 6; an = an–1 + 6 D. a1 = 6; an = 11 ● an–1 Example 2

B. Write a recursive formula for the sequence. 4, 18, 81, 364. 5, 1640 A. a1 = 4; an = an–1 + 4.5 B. a1 = 4; an = 4 ● an–1 + 14 C. a1 = 4; an = an–1 + 14 D. a1 = 4; an = 4.5 ● an–1 Example 2

C. Write a recursive formula for the sequence. a4 = 216 and r = 3 A. a1 = 8; an = an–1 + 3 B. a1 = 24; an = 3an–1 C. a1 = 8; an = 3an–1 D. a1 = 24; an = 8an–1 Example 2

x1 = f(x0) Iterate the function. = f(5) x0 = 5 Iterate a Function Find the first three iterates x1, x2, and x3 of the function f(x) = 3x – 1 for an initial value of x0 = 5. To find the first iterate x1, find the value of the function when x0 = 5. x1 = f(x0) Iterate the function. = f(5) x0 = 5 = 3(5) – 1 or 14 Simplify. To find the second iterate x2, substitute x1 for x. x2 = f(x1) Iterate the function. = f(14) x1 = 14 = 3(14) – 1 or 41 Simplify. Example 4

Substitute x2 for x to find the third iterate. Iterate a Function Substitute x2 for x to find the third iterate. x3 = f(x2) Iterate the function. = f(41) x2 = 41 = 3(41) – 1 or 122 Simplify. Answer: The first three iterates are 14, 41, and 122. Example 4

Find the first three iterates x1, x2, and x3 of the function f(x) = 2x + 1 for an initial value of x0 = 2. A. 3, 5, 11 B. 2, 5, 11 C. 5, 11, 23 D. 11, 23, 47 Example 4

Homework P. 695 # 12 – 42 (x3)

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