Write the sum $LaTeX: \displaystyle 1-1-3 \ldots -77-79$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-1-(1)=-2$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 1+(n-1)(-2)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 1+(n-1)(-2)=-79 \implies n = 41$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{41} \left(3 - 2 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 41(1-79) }{2}=-1599$ .