How to prove that if $n>2$ then $n!>n^{n/2}$ using induction?
2026-03-28 03:33:09.1774668789
Proving that if $n>2$ then $n!>n^{n/2}$ using induction.
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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 $n>2$.