I'm having trouble completing this induction proof, if someone could guide me on the right path, help is greatly appreciated.
Let $a_0,a_1,a_2,a_3,...$ be a sequence of integers such that
$a_0 = 3$
$a_1 = 10$
$a_2 = 38$
$a_{n+3} = 30 a_n - 31 a_{n+1} + 10 a_{n+2}$ for all $n \ge 0$
How can i prove that $a_n = 2^n + 3^n + 5^n$ ?
I know that I have to use complete induction here, but I'm not sure how to apply it.
$$ a_{n+1} = 10 a_n - 31 a_{n-1} + 30 a_{n-2} $$
$$ = 10(2^n+3^n+5^n) \\- 31(2^{n-1}+3^{n-1}+5^{n-1}) \\+ 30(2^{n-2}+3^{n-2}+5^{n-2})$$
$$ =2^{n-2}(40-62+30) \\+3^{n-2}(90-93+30) \\+5^{n-2}(250-155+30) $$
$$ = 2^{n+1}+3^{n+1}+5^{n+1}$$