Write the sum $LaTeX: \displaystyle -37-42-47 \ldots -247-252$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-42-(-37)=-5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -37+(n-1)(-5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -37+(n-1)(-5)=-252 \implies n = 44$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{44} \left(- 5 n - 32\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 44(-37-252) }{2}=-6358$ .