Write the sum $LaTeX: \displaystyle -16-20-24 \ldots -112-116$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-20-(-16)=-4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -16+(n-1)(-4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -16+(n-1)(-4)=-116 \implies n = 26$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{26} \left(- 4 n - 12\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 26(-16-116) }{2}=-1716$ .