Here's the question:

Use induction to prove that for every integer $\displaystyle n \geq 4$, $\displaystyle 3^n > n^3$.

Firstly, I'm confused whether to use the simple induction or the complete induction for this problem. I will try the simple one:

Base case: let $\displaystyle n \in N$ and let P(n) be the statement:

"$\displaystyle n \geq 4$, $\displaystyle 3^n > n^3$"

Since $\displaystyle n \geq 4$ I can't use P(1) as the base case so I'll do P(4) in its stead:

$\displaystyle 3^4 > 4^3 \Rightarrow 81>64$

Inductive step:

Suppose $\displaystyle k \geq 4$ and P(k) is true. So $\displaystyle 3^k > k^3$

Now we consider P(k+1)

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

$\displaystyle LHS = 3^k .3 > k^3 +1 = RHS $

What else can be done here to prove the inequality?

any help is appreciated