jmc

Question

Consider the sequence defined recursively by u_1 = a > 0 and u_n + 1 = -1u_n + 1for n 1. Express u_16 in terms of a.

Accepted Answer

We have that align* u_2 &= -1a + 1, u_3 &= -1-1a + 1 + 1 = -a + 1a, u_4 &= -1-a + 1a + 1 = a. align*Since u_4 = u_1, and each term depends only on the previous term, the sequence becomes periodic, with a period of length 3. Hence, u_16 = u_1 = a.