Write the sum $LaTeX: \displaystyle -9-17-25 \ldots -257-265$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-17-(-9)=-8$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -9+(n-1)(-8)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -9+(n-1)(-8)=-265 \implies n = 33$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{33} \left(- 8 n - 1\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 33(-9-265) }{2}=-4521$ .