Question

13. Consider the sequence of numbers ao, ai, a2, a3, given by ao-2, ai-3, and for any positive integer k 2, a3ak 2ak-1. (a) Evaluate a2,a3, a4,as. Show your work. (b) Prove that for all positive integers n, an 2 +1

Answer 1

(a) a_0 = 2, a_1 = 3 a_k + 1 = 3a_k - 2a_k - 1 for k greaterthanorequalto 2 (1) therefore a_2 = 3a_1 - 2a_0 = 3(3) - 2(2) = 5 a_3 = 3(5) - 2(3) = 15 - 6 = 9 a_4 = 3(9) - 2(5) = 27 - 10 = 17 a_5 = 3(17) - 2(4) = 51 - 18 = 33 (b) to prove a_n = 2^n + 1 for n element of N basis of induction for n = 1 a_1 = 2 + 1 3 = 3 is true inductive step let given statement is true for n = 1, 2, ..., k - 1, k therefore a_k = 2^k + 1 a_k - 1 = 2^k - 1 + 1 (2) now, we prove the statement for n = k + 1 that a_k + 1 = 2^k + 1 + 1 taking a_k + 1 = 3a_k - 2a_k + 1(using) a_k + 1 = 3(2^k + 1) - 2(2^k - 1 + 1)(using(2)) = 3(2^k) + 3 - 2(2^k - 1) - 2