Inequality of positive integers

Question

I have to prove the following inequalities for positive integer $n$: $$n^n\geqslant\left(\frac{n+1}2\right)^{n+1}$$ $$ 2^ {n(n+1)}> (n+1)^{n+1} \left(\frac{n}{1}\right)^{n} \left(\frac{n-1}{2}\right)^{n-1}...\left(\frac{2}{n-1}\right)^{2} \left(\frac{1}{n}\right) $$ I tried using weighted $AM > GM > HM $ for $ (n+1), \left(\frac{n}{1}\right), \left(\frac{n-1}{2}\right),...\left(\frac{2}{n-1}\right), \left(\frac{1}{n}\right)$ . Any sort of hint/help is appreciated. Thank you.