Use mathematical induction to prove that $(3^n+7^n)-2$ is divisible by 8 for all non-negative integers. Base step: $3^0 + 7^0 - 2 = 0$ and $8|0$
Suppose that $8|f(n)$, let's say $f(n)= (3^n+7^n)-2=  8k$
Then $f(n+1) = (3^{n+1}+7^{n+1})-2$
$(3*3^{n}+7*7^{n})-2$
This is the part I get stuck. Any help would be really appreciated.
Thanks.

$f(n)$ satisfies $f(n) = 11 f(n-1) - 31f(n-2) +21f(n-3)$
So by induction it's enough to check this is true for $n =0,1,2$.
