Prove that $13\vert(3^{n+1} +3^{n} +3^{n-1})$ 
Prove that $3^{n+1} +3^{n} +3^{n-1}$ is divisible by $13$ for all positive integral values of $n$


Observe that
$$1\bmod13=1,\\3\bmod13=3,\\9\bmod13=9,\\27\bmod13=1,\\81\bmod13=3,\\243\bmod13=9,\\...$$
The modulos repeat periodically and the sum on a period is $13$.
