Find all $n$ such that $3^{2n+1}+2^{n+2}$ is divisible by $7$ Find all $n$ such that 
$3^{2n+1}+2^{n+2}$ is divisible by $7$
Prove that your answer is correct 
So I am not allowed to use mods,  as is a calculus question, I have tried by induction but can't get to prove that it works for $k+1$, by multiplying the equation by powers of $2$ and $3$. 
Thank you for your help 

Note that 
$$\begin{split}
3^{2(k+1)+1} + 2^{(k+1)+2} &= 9\cdot 3^{2k+1} + 2\cdot 2^{k+2}\\
&= 7 \cdot 3^{2k+1} + 2 (3^{2k+1} + 2^{k+2}). 
\end{split}$$
