Write the sum $LaTeX: \displaystyle -37-46-55 \ldots -271-280$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-46-(-37)=-9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -37+(n-1)(-9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -37+(n-1)(-9)=-280 \implies n = 28$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{28} \left(- 9 n - 28\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 28(-37-280) }{2}=-4438$ .