Write the sum $LaTeX: \displaystyle -20-29-38 \ldots -407-416$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-29-(-20)=-9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -20+(n-1)(-9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -20+(n-1)(-9)=-416 \implies n = 45$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{45} \left(- 9 n - 11\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 45(-20-416) }{2}=-9810$ .