\begin{align*} \sigma^2 & = \sum_{x=1}^n x^2 \cdot \frac{1}{n} - \mu^2\\ & = \frac{1}{n}\sum_{x=1}^n x^2 - \left ( \frac{1+n}{2}\right )^2 \\ & = \frac{1}{n} \frac{n(n+1)(2n+1)}{6} - \frac{1+2n+n^2}{4}\\ & = \frac{(2n^2+3n+1)}{6} - \frac{1+2n+n^2}{4}\\ & = \frac{(4n^2+6n+2)}{12} - \frac{3+6n+3n^2}{12}\\ & = \frac{(n^2-1)}{12} \end{align*}