Write the sum $LaTeX: \displaystyle 22+18+14 \ldots -98-102$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=18-(22)=-4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 22+(n-1)(-4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 22+(n-1)(-4)=-102 \implies n = 32$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{32} \left(26 - 4 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 32(22-102) }{2}=-1280$ .