Write the sum $LaTeX: \displaystyle 15+13+11 \ldots -67-69$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=13-(15)=-2$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 15+(n-1)(-2)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 15+(n-1)(-2)=-69 \implies n = 43$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{43} \left(17 - 2 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 43(15-69) }{2}=-1161$ .