Prove that 3^n > n^3 for all positive integers n.

I first tried n= 1, which showed 3>1

n= 2, 9>8

but n = 3 became 27 = 27, which is not true, but everything else works afterwards.

I was given the hint to use induction. So I tried this:

3^(n+1) > (n+1)^3

3*(3^n) > n^3 + 3n^2 + 3n + 1

What do I do next?