Prove $3^n > n^2$ by induction Prove that $$3^n > n^2$$
I am using induction and I understand that when $n=1$ it is true. The induction hypothesis is when $n=k$ so $3^k>k^2$. So for the induction step we have $n=k+1$ so 
$3^{k+1} > (k+1)^2$ which is equal to $3\cdot3^k > k^2+2k+1$. I know you multiple both sides of the induction hypothesis by $3$ but I'm not sure what to do next.

You start with knowing that $3^k > k^2$ and you prove that if you know that then you prove that $3^{k+1} > (k+1)^2$.
One straightforward and Doy! way is to note:
$3^k > k^2$
$3^k + 3^k + 3^k> k^2 + k^2+k^2$.  
Since $k \ge 2$ then $k^2 \ge 2k$ and as $k \ge 2$ then $k^2 > 1$
So 
$3^k + 3^k + 3^k > k^2 + 2k + 1$
$3^{k+1} = 3*3^k > (k+1)^2$.
