jmc

Question

Given a sequence a_1, a_2, a_3, , let S_n denote the sum of the first n terms of the sequence. If a_1 = 1 and a_n = 2S_n^22S_n - 1for all n 2, then find a_100.

Accepted Answer

By definition of S_n, we can write a_n = S_n - S_n - 1. Then S_n - S_n - 1 = 2S_n^22S_n - 1,so (2S_n - 1)(S_n - S_n - 1) = 2S_n^2. This simplifies to S_n - 1 = 2S_n - 1 S_n + S_n.If S_n = 0, then S_n - 1 = 0. This tells us that if S_n = 0, then all previous sums must be equal to 0 as well. Since S_1 = 1, we conclude that all the S_n are nonzero. Thus, we can divide both sides by S_n - 1 S_n, to get 1S_n = 1S_n - 1 + 2.Since 1S_1 = 1, it follows that 1S_2 = 3, 1S_3 = 5, and so on. In general, 1S_n = 2n - 1,so S_n = 12n - 1. Therefore, a_100 = S_100 - S_99 = 1199 - 1197 = -239203.