Application of AM >= GM solution

The problem was to show that

$$\left(1 + \frac{1}{n+1}\right)^{n+1} \gt \left(1 + \frac{1}{n}\right)^{n}$$

using the AM >= GM inequality.

Let $x_1 = x_2 = \dots = x_{n} = 1 + \frac{1}{n}$ and $x_{n+1} = 1$

Applying the AM >= GM to the $x_i$ we get that

$$\frac{x_1 + x_2 + \dots + x_{n+1}}{n+1} \ge \sqrt[n+1]{x_1x_2\dots x_{n+1}}$$

i.e

$$ \frac{n\left(1+\frac{1}{n}\right) + 1}{n+1} \ge \sqrt[n+1]{\left(1 + \frac{1}{n}\right)^{n}}$$

i.e

$$ \left(1+\frac{1}{n+1}\right)  \ge \left(1 + \frac{1}{n}\right)^{\frac{n}{n+1}}$$

Raising both sides to the $(n+1)^{th}$ power gives us the result.