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EXAMPLE 1

EXAMPLE 1. Evaluate recursive rules. Write the first six terms of the sequence. a. a 0 = 1, a n = a n – 1 + 4. b. a 1 = 1, a n = 3 a n – 1. SOLUTION. a. a 0 = 1. b. a 1 = 1. a 1 = a 0 + 4 = 1 + 4 = 5. a 2 = 3 a 1 = 3(1) = 3. a 2 = a 1 + 4 = 5 + 4 = 9.

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EXAMPLE 1

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  1. EXAMPLE 1 Evaluate recursive rules Write the first six terms of the sequence. a. a0 = 1, an= an – 1 + 4 b. a1 = 1, an= 3an – 1 SOLUTION a. a0 = 1 b. a1 = 1 a1 = a0 + 4 = 1 + 4 = 5 a2 = 3a1 = 3(1) = 3 a2 = a1 + 4 = 5 + 4 = 9 a3 = 3a2 = 3(3) = 9 a3 = a2 + 4 = 9 + 4 = 13 a4 = 3a3 = 3(9) = 27 a5 = 3a4 = 3(27) = 81 a4 = a3 + 4 = 13 + 4 = 17 a5 = a4 + 4 = 17 + 4 = 21 a6 = 3a5 = 3(81) = 243

  2. ANSWER So, a recursive rule for the sequence isa1 = 3, an= an– 1 + 10. EXAMPLE 2 Write recursive rules Write the first six terms of the sequence. a. 3, 13, 23, 33, 43, . . . b. 16, 40, 100, 250, 625, . . . SOLUTION The sequence is arithmetic with first term a1 = 3 and common difference d = 13 – 3 = 10. an= an – 1 + d General recursive equation for an = an – 1 + 10 Substitute 10 for d.

  3. b. The sequence is geometric with first term a1 = 16 and common ratio r = = 2.5. an= ran– 1 40 16 ANSWER So, a recursive rule for the sequence is a1 = 16,an= 2.5an – 1. EXAMPLE 2 Write recursive rules General recursive equation for an = 2.5an – 1 Substitute 2.5 for r.

  4. – 1. a1 = 3, an= an – 1 7 ANSWER ANSWER ANSWER 3, –4, –11, –18, –25 162, 81, 40.5, 20.25, 10.125 1, 2, 4, 7, 11 for Examples 1 and 2 GUIDED PRACTICE Write the first five terms of the sequence. 2. a0 = 162, an= 0.5an – 1 3. a0 = 1, an= an – 1 + n

  5. ANSWER So, a recursive rule for the sequence is a1 = 2, an= 7an – 1 for Examples 1 and 2 GUIDED PRACTICE Write a recursive rule for the sequence. 5. 2, 14, 98, 686, 4802, . . . 6. 19, 13, 7, 1, – 5, . . . ANSWER So, a recursive rule for the sequence is a1 = 19, and an= an – 1 – 6.

  6. 1 a = 324, and an= an – 1 3 for Examples 1 and 2 GUIDED PRACTICE Write a recursive rule for the sequence. 7. 11, 22, 33, 44, 55, . . . ANSWER So, a recursive rule for the sequence is a1 = 11, and an= an – 1 + 11. 8. 324, 108, 36, 12, 4, . . . ANSWER So, a recursive rule for the sequence is

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