Write the sum $LaTeX: \displaystyle -31-22-13 \ldots +329+338$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-22-(-31)=9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -31+(n-1)(9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -31+(n-1)(9)=338 \implies n = 42$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{42} \left(9 n - 40\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 42(-31+338) }{2}=6447$ .