Write the sum $LaTeX: \displaystyle 8+1-6 \ldots -139-146$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=1-(8)=-7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 8+(n-1)(-7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 8+(n-1)(-7)=-146 \implies n = 23$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{23} \left(15 - 7 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 23(8-146) }{2}=-1587$ .