Write the sum $LaTeX: \displaystyle 9+0-9 \ldots -207-216$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=0-(9)=-9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 9+(n-1)(-9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 9+(n-1)(-9)=-216 \implies n = 26$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{26} \left(18 - 9 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 26(9-216) }{2}=-2691$ .