Write the sum $LaTeX: \displaystyle -31-37-43 \ldots -205-211$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-37-(-31)=-6$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -31+(n-1)(-6)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -31+(n-1)(-6)=-211 \implies n = 31$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{31} \left(- 6 n - 25\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 31(-31-211) }{2}=-3751$ .