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 $n$


Besides the obvious factorization approach, you can reason by induction.
$$3^{(n+1)+1}+3^{(n+1)}+3^{(n+1-1)}=3\cdot 3^{n+1}+3\cdot 3^n+3\cdot 3^{n-1}=3 \cdot (3^{n+1}+3^n+3^{n-1}),$$
and
$$3^{1+1}+3^{1}+3^{1-1}=13.$$
Hence
$$S_{n+1}=3\cdot S_n\text{, and }S_1=13.$$
