Write the sum $LaTeX: \displaystyle 18+17+16 \ldots -13-14$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=17-(18)=-1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 18+(n-1)(-1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 18+(n-1)(-1)=-14 \implies n = 33$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{33} \left(19 - n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 33(18-14) }{2}=66$ .