Write the sum $LaTeX: \displaystyle 36+31+26 \ldots -79-84$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=31-(36)=-5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 36+(n-1)(-5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 36+(n-1)(-5)=-84 \implies n = 25$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{25} \left(41 - 5 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 25(36-84) }{2}=-600$ .