Prove that for any positive integer $n$, the arithmetic mean of $\sqrt[1]{1},\sqrt[2]{2},\sqrt[3]{3},\ldots ,\sqrt[n]{n}$ lies in $\left[ 1,1+\frac{2\sqrt{2}}{\sqrt{n}} \right]$ .
Source: SRMC 2012
Tags: algebra, inequalities
Prove that for any positive integer $n$, the arithmetic mean of $\sqrt[1]{1},\sqrt[2]{2},\sqrt[3]{3},\ldots ,\sqrt[n]{n}$ lies in $\left[ 1,1+\frac{2\sqrt{2}}{\sqrt{n}} \right]$ .
