For what natural $n$ does $3^n > n^3$ hold true? Prove by induction 
For what natural $n$ does $3^n > n^3$ hold true? 

I figured that it holds true for all $n$ except $n = 3$. I am not sure how to prove it by induction. I proved it by $p(k) \implies p(k+1)$ but that doesn't show that $n \neq 3$ 

The very final step would be:
\begin{align*}
3^{n+1}&=3\cdot 3^{n}\\
&> 3n^{3}
\end{align*} and we have 
\begin{align*}
3n^{3}-(n+1)^{3}&=2n^{3}-3n^{2}-3n-1\\
&=n^{3}-3n^{2}+n^{3}-3n-1\\
&\geq n^{2}(n-3)+n^{3}-3n-n\\
&=n^{2}(n-3)+n(n+2)(n-2)\\
&>3\cdot5\cdot 1\\
&>0
\end{align*}
since $n> 3$. As a matter of simple checking, one should starts from $4$.
