Write the sum $LaTeX: \displaystyle -41-35-29 \ldots +133+139$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-35-(-41)=6$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -41+(n-1)(6)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -41+(n-1)(6)=139 \implies n = 31$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{31} \left(6 n - 47\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 31(-41+139) }{2}=1519$ .