Write the sum $LaTeX: \displaystyle 13+12+11 \ldots -30-31$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=12-(13)=-1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 13+(n-1)(-1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 13+(n-1)(-1)=-31 \implies n = 45$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{45} \left(14 - n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 45(13-31) }{2}=-405$ .