Write the sum $LaTeX: \displaystyle 4-5-14 \ldots -266-275$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-5-(4)=-9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 4+(n-1)(-9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 4+(n-1)(-9)=-275 \implies n = 32$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{32} \left(13 - 9 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 32(4-275) }{2}=-4336$ .