Write the sum $LaTeX: \displaystyle 31+24+17 \ldots -235-242$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=24-(31)=-7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 31+(n-1)(-7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 31+(n-1)(-7)=-242 \implies n = 40$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{40} \left(38 - 7 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 40(31-242) }{2}=-4220$ .