Answer
$α_n=2^n$
Work Step by Step
The characteristic equation of this recurrence relation is
$$r^4 −5r^2 + 4 = 0$$
Because $r^4 −5r^2 + 4 = 0=(r^2 − 4)(r^2 − 1) = 0$.
It follows that the solutions are $r_1 = 1, r_2 = -1, r_3=2, r_4=-2$
By Theorem 4 the solutions of this recurrence relation are of the form $α_n= α_1(-1)^n + α_21^n + α_3(-2)^n + α_42^n$
The simultaneous solutions are $α_1 =1, α_2 =1, α_3 = 0, α_4=1$
Therefore, $a_n = −1^n +1^n + 2^n = 2^n$