$ \sum_{k=1}^n k $ is a arithmetic series because it has a common difference of 1.
The general sum of an arithmetic series is $ n \frac {(a_1+a_n)} {2} $ where $ a_1 $ is the first term and $ a_n $ the last.
This is how Brian did his simplification.
$ (\frac{1}{n}) \sum_{k=1}^n k= \frac{1}{n} (\frac{n(n+1)}{2}) $
