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.
Solution:
equivalently \(\displaystyle \sqrt[n+2]{\left(\frac{n}{n+1}\right)^{n+1}\cdot 1}{<} \frac{(n+1)\left(\frac{n}{n+1}\right)+1}{n+2}\).