Write the sum $LaTeX: \displaystyle -17-23-29 \ldots -305-311$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-23-(-17)=-6$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -17+(n-1)(-6)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -17+(n-1)(-6)=-311 \implies n = 50$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{50} \left(- 6 n - 11\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 50(-17-311) }{2}=-8200$ .