Question

Solve and show work for problem 8

Problem 8. Consider the sequence defined by ao = 1, ai-3, and a',--2an-i-an-2 for n Use the generating function for this sequence to find an explicit (closed) formula for a 2.

Answer 1

Given recurrence relation: - a_n = -2a_n + 1 implies a_n + 2a_n-1 + a_n-2 = 0 and hence, the characteristic equation is: r^2 + 2r + 1 = 0 implies (r + 1)^2 = 0 implies r = -1, -1 implies solution: a_n = c_1 r^n_1 + c_2 n r^n_2 (therefore r_1 = r_2) implies a_n = (c_1 + c_2 n) (-1)^n a_0 = 1 a_1 = 3} implies a_0 = c_1 = 1: a_1 = (c_1 + c_2) (-1) = 3 implies 1 + C_2 = -3 implies c_2 = -4 implies a_n = (1 -4 n) (-1)^n