Prove by mathematical induction that 3^(3n+1) + 2^(n+1) is divisible by 5

check for n=0

3+2=5

Now assume that it's true for n=k:
3^(3k+1) + 2^(k+1) = 5n

When n=k+1, then,
3^(3k+4) + 2^(k+2)
= 27*3^(3k+1) + 2*2^(k+1)
= 5n + 26*3^(3k+1) + 2^(k+1)
= 5n + 3^(3k+1) + 2^(k+1) + 25*3^(3k+1)
= 5n+5n+25*3^(3k+1)
But 25 is a multiple of 5, so the sum is also.

So, if true for n=k, then true for n=k+1.
true for n=0, so true for n=1,2,3...