Write the sum $LaTeX: \displaystyle 32+29+26 \ldots -58-61$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=29-(32)=-3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 32+(n-1)(-3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 32+(n-1)(-3)=-61 \implies n = 32$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{32} \left(35 - 3 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 32(32-61) }{2}=-464$ .