Write the sum $LaTeX: \displaystyle -46-40-34 \ldots +134+140$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-40-(-46)=6$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -46+(n-1)(6)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -46+(n-1)(6)=140 \implies n = 32$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{32} \left(6 n - 52\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 32(-46+140) }{2}=1504$ .