Problem 335 (difficulty: 5/10)

Prove that

\(\displaystyle \left (1+\frac{1}{n} \right )^{n+1}>\left (1+\frac{1}{n+1} \right)^{n+2},\)

in other words the sequence \(\displaystyle a_n=\left (1+\frac{1}{n} \right )^{n+1}\) is strictly monotone decreasing.

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