EXAMPLE 1 Evaluate recursive rules Write the first six terms of the sequence. a. a 0 = 1, a n = a n – b. a 1 = 1, a n = 3a n – 1 SOLUTION a. a 0 = 1 a 1 = a = = 5 a 2 = a = = 9 a 3 = a = = 13 a 4 = a = = 17 a 5 = a = = 21 b. a 1 = 1 a 2 = 3a 1 = 3(1) = 3 a 3 = 3a 2 = 3(3) = 9 a 4 = 3a 3 = 3(9) = 27 a 5 = 3a 4 = 3(27) = 81 a 6 = 3a 5 = 3(81) = 243

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

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

GUIDED PRACTICE for Examples 1 and 2 Write the first five terms of the sequence. 1. a 1 = 3, a n = a n – 1 7 – ANSWER 3, –4, –11, –18, –25 2. a 0 = 162, a n = 0.5a n – 1 ANSWER 162, 81, 40.5, 20.25, a 0 = 1, a n = a n – 1 + n ANSWER 1, 2, 4, 7, 11

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

GUIDED PRACTICE for Examples 1 and 2 Write a recursive rule for the sequence , 22, 33, 44, 55,... So, a recursive rule for the sequence isANSWER a 1 = 11, and a n = a n – , 108, 36, 12, 4,... So, a recursive rule for the sequence isANSWER a = 324, and a n = a n – 1 1 3