Prove that $3^{n+1}+3^n+3^{n-1}$ is divisible by $13$. Prove that $3^{n+1}+3^n+3^{n-1}$ is divisible by $13$ for all positive integral values of $n$.
I tried: $3^n \cdot 3^1+3^n+3^n\cdot\frac{1}{3}$
Then what should I do next? Help please?

I'm not sure how I would proceed from your path, but here is an alternative:
$$3^{n+1} + 3^n + 3^{n-1}= 3^{n-1}(3^2+3+1)=3^{n-1} \cdot 13$$
So:
$$\frac{3^{n+1} + 3^n + 3^{n-1}}{13}=3^{n-1}$$
And since $n \in \mathbb{N}$:
$$13|(3^{n+1} + 3^n + 3^{n-1})$$
