Write the sum $LaTeX: \displaystyle -9-13-17 \ldots -181-185$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-13-(-9)=-4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -9+(n-1)(-4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -9+(n-1)(-4)=-185 \implies n = 45$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{45} \left(- 4 n - 5\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 45(-9-185) }{2}=-4365$ .