Write the sum $LaTeX: \displaystyle -7-11-15 \ldots -163-167$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-11-(-7)=-4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -7+(n-1)(-4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -7+(n-1)(-4)=-167 \implies n = 41$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{41} \left(- 4 n - 3\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 41(-7-167) }{2}=-3567$ .