Finite Sum

Problem:

$\displaystyle\sum_{z=0}^{2016}\left[\frac{z (z+1)}{2}\right]=\,?$

Solution:

$\displaystyle\sum_{z=0}^{2016}\left[\frac{z (z+1)}{2}\right]$

$=\frac{1}{2} \times \displaystyle\sum_{z=0}^{2016}\left[z^2 + z\right]$

$=\frac{1}{2} \times \left[ \frac{2016(2016+1)(2(2016)+1)}{6}\right] + \left[ \frac{2016(2016+1)}{2} \right]$

$=1367622816$

Conclusion:

$\displaystyle\sum_{z=0}^{2016}\left[\frac{z (z+1)}{2}\right]=\boxed{1367622816}$