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.

$3 (27^{n})+2 (2^{n}) $ is equal to your value,so $3 (2^{n})+2 (2^{n})=5( 2^{n}) $ has equal reminder (as dividing by 5)with the previous value because 27 has the same reminder ( with respect to 5) as 2.so it is proved.
