Prove that this number is divisible by 7 Without using induction, how can it be proved that 7 divides $3^{2n+1}+2^{n+2}$ for each $n\in\mathbb{N}$?
I tried to expand it using $\frac{x^{n+1}-1}{x-1}=1+x+..+x^n$ but I had no success.
It would be great if more than one proof is provided.

HINT: Simplify $3^{2n+1}+2^{n+2}$ modulo $7$, using the fact that $3^{2n+1}=3\cdot 3^{2n}$ and $3^2\equiv2\pmod7$.
