<s>[INST] 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 
 [/INST] $3^{2n+1}+2^{n+2}$ is divisible by $7$ for all $n$:
$
3^{2n+1}+2^{n+2}=\\
=3\cdot 9^n+4\cdot 2^n\\
=3\cdot (7+2)^n+(7-3)\cdot 2^n\\
=3(7a+2^n)+7\cdot2^n-3\cdot 2^n\\
=7(3a+2^n)
$
where I have used the binomial theorem for getting $(7+2)^n=7a+2^n$.
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