inequality - Prove through induction that $3^n > n^3$ for $n geq 4$

I'm new to induction and have not done induction with inequalities before, so I get stuck at proving after the 3rd step.

The question is:

Use induction to show that $3^n > n^3$ for $n \geq 4$.

I have so far:

Step 1:
Prove for $n=4$ (since question states this)
$3^4 > 4^3$

$81 > 64 $

which is true

Step 2:
Assume true for $n=k$

$3^k > k^3$

Step 3:
Prove for $n = k+1$

$3^{k+1} > (k+1)^3$

Here I expand to:

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

However I have no idea how to prove this.

Thanks for any help given

Answer

$3^{k+1}>3k^3$

We need $3^{k+1}>(k+1)^3 $

So, it sufficient to prove $3k^3>(k+1)^3\iff \left(1+\frac1k\right)^3<3$

For $k=3,\left(1+\frac1k\right)^3=\frac{64}{27}<3$

and $\left(1+\frac1{k+1}\right)^3<\left(1+\frac1k\right)^3$

$\implies \left(1+\frac1k\right)^3<3$ for $k\ge3$