Prove that $5$ divides $3^{3n+1}+2^{n+1}$ 
Prove that $5$ divides $3^{3n+1}+2^{n+1}$

I tried to prove the result by induction but I couldn't.
The result is true for $n=1$.
Suppose that the result is true for $n$ i.e $3^{3n+1}+2^{n+1}=5k$ for some $k\in \mathbb{N}$. We study the term 
$$3^{3n+4}+2^{n+2}=3^{3n+1}3^3+2^{n+1}2$$
I tried to prove that that the difference is a multiple of $5$.
$$3^{3n+1}3^3+2^{n+1}2-3^{3n+1}+2^{n+1}=2(3^{3n+1}\cdot 13+2^n)$$
Therefore it's enough to prove that $3^{3n+1}\cdot 13+2^n$ is a multiple of $5$. But if I do again this method applied to this "new problem" is get something similar. I think that there exist a different method to do this using induction.

Your approach works if you just subtract once more:
$$
3^{3n+4} + 2^{n+2} - 2\cdot 5k\\
= 27\cdot 3^{3n+1} + 2\cdot 2^{n+1} - 2(3^{3n+1} + 2^{n+1})\\
= 25\cdot 3^{3n+1}
$$
and you're done.
