Write the sum $LaTeX: \displaystyle -39-37-35 \ldots +21+23$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-37-(-39)=2$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -39+(n-1)(2)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -39+(n-1)(2)=23 \implies n = 32$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{32} \left(2 n - 41\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 32(-39+23) }{2}=-256$ .