Proving that if $n>2$ then $n!>n^{n/2}$ using induction.

Question

How to prove that if $n>2$ then $n!>n^{n/2}$ using induction?

Did you try the base case? Where are you struggling in the inductive case? :-) — Hakim, Sep 08 '14 at 17:53
I get (k+1)!>k^(k/2)*(k+1) and don't know what I can do then. — Sona, Sep 08 '14 at 17:56

score 4 · Accepted Answer · answered Sep 08 '14 at 18:07

4

Hint:

$\left(n+1\right)!=n!\left(n+1\right)>n^{\frac{n}{2}}\left(n+1\right)$.

So it is enough to prove that: $$n^{\frac{n}{2}}\left(n+1\right)\geq\left(n+1\right)^{\frac{n+1}{2}}$$ or equivalently: $$n+1\geq\left(1+\frac{1}{n}\right)^{n}$$ This for $n>2$.

answered Sep 08 '14 at 18:07

drhab

151,093

How do you get n+1≥(1+1/n)^n – Sona Sep 08 '14 at 18:14
1
on both sides take the square. 2) on both sides divide by $n+1$.
on both sides divide by $n^{n}$

drhab

Sep 08 '14 at 18:19

@Sona You need $n^{n/2}\geq (n+1)^{(n-1)/2}$. – zibadawa timmy Sep 08 '14 at 18:19

Thank you very much for help. I'm at my first year at university and couldn't do my homework. – Sona Sep 08 '14 at 18:24

2

You are very welcome. Persevere and good luck with your study. – drhab Sep 08 '14 at 18:25

Proving that if $n>2$ then $n!>n^{n/2}$ using induction.

1 Answers1