How can I prove this by mathematical induction

Question

$n!>n^{n/2}$. For every positive integer greater than $2$

How much do you know about mathematical induction? Can you prove the basis, for example? — AloneAndConfused, May 07 '16 at 15:59
Possible duplicate of How to prove this inequality with factorials: $n!>n^{\frac {n}{2}}$ — , May 07 '16 at 19:56

score 4 · Answer 1 · answered May 07 '16 at 16:04

Start with $$n^{\frac{n}{2}} <n!$$ multiply by $n+1$ to get $$(n+1)n^{\frac{n}{2}} <(n+1)!$$

we now would like to show that

$$(n+1)^{\frac{n+1}{2}} \leq (n+1)n^{\frac{n}{2}} $$ If we square this and rearrange we get

$$\left(1+\frac{1}{n}\right)^n\leq n+1$$ However $\left(1+\frac{1}{n}\right)^n\leq e<3$ is well known.

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