Write the sum $LaTeX: \displaystyle -27-36-45 \ldots -333-342$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-36-(-27)=-9$ . 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 $LaTeX: \displaystyle -27+(n-1)(-9)=-342 \implies n = 36$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{36} \left(- 9 n - 18\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 36(-27-342) }{2}=-6642$ .