Mathematical induction with an inequality and factorial notation: $n! > n^2$

Question

I'm having difficulty proving $n! > n^2$ for $n \ge 4$ I have previously solved a similar problem but it is $n! > 2^n$. Now I don't know how to solve this. I have only come as far as solving for the base case. Thank you so much for helping a struggling student out.

Answer 1

So the problem is $$n!>n^2 \space \forall n\ge4$$

We'll try to tackle it using induction:

Base case: $$4!>16$$

Hypothesis: $$n!>n^2$$

Induction: $$(n+1)!>(n+1)^2 \Rightarrow (n+1)(n)!\gt(n+1)^2 \Rightarrow n!\gt\frac{(n+1)^2}{(n+1)} \Rightarrow \bbox[border:1px solid red]{n!\gt n+1}$$

The induction is proved.

Answer 2

Hint: Try to justify \begin{align} n(n^2-2n+1) \ge n^2. \end{align} for $n\geq 4$.