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.
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I have solved this question, but how does it help here if you can please elaborate? – Sharmi C May 08 '22 at 15:36
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In the inequality $a^ab^b\geqslant\left(\frac{a+b}2\right)^{a+b}$, if you take $a=n$ and $b=1$, it becomes $n^n\geqslant\left(\frac{n+1}2\right)^{n+1}$. – José Carlos Santos May 08 '22 at 15:52
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Could you provide a hint for 2nd one? – Sharmi C May 08 '22 at 15:53
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You shall find an answer here. – José Carlos Santos May 08 '22 at 15:57