Write the sum $LaTeX: \displaystyle 27+18+9 \ldots -279-288$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=18-(27)=-9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 27+(n-1)(-9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 27+(n-1)(-9)=-288 \implies n = 36$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{36} \left(36 - 9 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 36(27-288) }{2}=-4698$ .