Write the sum $LaTeX: \displaystyle 7+3-1 \ldots -77-81$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=3-(7)=-4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 7+(n-1)(-4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 7+(n-1)(-4)=-81 \implies n = 23$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{23} \left(11 - 4 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 23(7-81) }{2}=-851$ .