induction help

For the base case: n = 0.

$\text{When }n = 0,\\ 5^{3^{0}} + 1 = 6\\ 3^{0 + 1} = 3\\ 3|6$

Assume $5^{3^k} + 1\text{ is divisible }3^{k + 1}$.

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

Now, $5^{2(3^k)} \equiv 2^{2(3^k)} \equiv 4^{3^k} \equiv 1^{3^k} \equiv 1 \pmod 3$

$5^{3^k}\equiv 2^{3^k}\equiv 2^{2K + 1}\equiv 2 \pmod 3$ for some $K\in \mathbb Z$

Therefore $3|\left(5^{2(3^k)} - 5^{3^k} + 1\right)$

The proposition is proved.

P.S.: I cheated and used some modular arithmetic ^_^