Write the sum $LaTeX: \displaystyle 26+22+18 \ldots -90-94$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=22-(26)=-4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 26+(n-1)(-4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 26+(n-1)(-4)=-94 \implies n = 31$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{31} \left(30 - 4 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 31(26-94) }{2}=-1054$ .