Write the sum $LaTeX: \displaystyle -28-33-38 \ldots -158-163$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-33-(-28)=-5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -28+(n-1)(-5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -28+(n-1)(-5)=-163 \implies n = 28$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{28} \left(- 5 n - 23\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 28(-28-163) }{2}=-2674$ .