<s>[INST] Prove by induction that $1^1\cdot 2^2\cdot \dots\cdot n^n\leq \left(\frac{2n+1}{3}\right)^{\frac{n(n+1)}{2}}.$ How can I prove by induction that for every natural number $n$, 
$$1^1\cdot 2^2\cdot \dots\cdot n^n\leq \left(\frac{2n+1}{3}\right)^{\frac{n(n+1)}{2}}.$$
 [/INST] Hint. Instead of induction, I warmly recommend the use of the AGM inequality. Note that
$$\left(1^1\cdot 2^2\cdot \dots\cdot n^n\right)^{\frac{2}{n(n+1)}}$$
is the geometric mean of the numbers
$$1,2,2,3,3,3,\dots,\underbrace{n,\dots,n}_{\text{$n$ times}}.$$
What is their arithmetic mean?
$$\frac{1+2+2+3+3+3+\dots+\overbrace{n+\dots+n}^{\text{$n$ times}}}{n(n+1)/2}=\frac{1+2^2+3^2+\dots+n^2}{n(n+1)/2}=?$$
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