Answer
$a_n = −2^n − 2(−1)^n + 6$
Work Step by Step
The characteristic equation of this recurrence relation is
$$r^3 −2r^2 −r+ 2 = 0$$
Because $r^3 −2r^2 −r+ 2 = 0=r^2(r − 2) − (r − 2)$, so $(r − 2)(r^2 − 1) = 0$.
It follows that the solutions are $r_1 = 2, r_2 = −1, r_3 = 1$
By Theorem 4 the solutions of this recurrence relation are of the form $α_n= α_12^n + α_2(−1)^n + α_3$
The simultaneous solutions are $α_1 = −1, α_2 = −2, α_3 = 6$
Therefore, $a_n = −2^n − 2(−1)^n + 6$
