Write the sum $LaTeX: \displaystyle 32+25+18 \ldots -143-150$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=25-(32)=-7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 32+(n-1)(-7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 32+(n-1)(-7)=-150 \implies n = 27$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{27} \left(39 - 7 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 27(32-150) }{2}=-1593$ .