Proving by induction: $2^n > n^3 $ for any natural number $n > 9$ I need to prove that $$ 2^n > n^3\quad \forall n\in \mathbb N, \;n>9.$$
Now that is actually very easy if we prove it for real numbers using calculus. But I need a proof that uses mathematical induction. 
I tried the problem for a long time, but got stuck at one step - I have to prove that:

$$ k^3 > 3k^2 + 3k + 1 $$

Hints???

Here's another way. Suppose $k>9$. Then:
$$ \begin{align*}
k^3&=(k)k^2\\
&>9k^2\\
&=3k^2+3k^2+3k^2\\
&=3k^2+3(k)k+3(k)^2\\
&>3k^2+3(9)k+3(9)^2\\
&=3k^2+27k+243\\
&>3k^2+3k+1\\
\end{align*}$$
