Write the sum $LaTeX: \displaystyle -45-48-51 \ldots -138-141$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-48-(-45)=-3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -45+(n-1)(-3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -45+(n-1)(-3)=-141 \implies n = 33$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{33} \left(- 3 n - 42\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 33(-45-141) }{2}=-3069$ .