Write the sum $LaTeX: \displaystyle 32+27+22 \ldots -168-173$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=27-(32)=-5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 32+(n-1)(-5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 32+(n-1)(-5)=-173 \implies n = 42$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{42} \left(37 - 5 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 42(32-173) }{2}=-2961$ .