Prove that $5^{3^n} + 1$ is divisible by $3^{n + 1}$ for all nonnegative integers $n.$

Question

Prove that $5^{3^n} + 1$ is divisible by $3^{n + 1}$ for all nonnegative integers $n.$

I tried to use Proof by Induction, but I'm stuck on the case for when $n=k+1.$

Answer 1

Hint:

$5^{3^{k+1}}+1=(5^{3^k})^3+1=(5^{3^k}+1)(5^{2\cdot3^k}-5^{3^k}+1)$