Write the sum $LaTeX: \displaystyle -20-26-32 \ldots -272-278$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-26-(-20)=-6$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -20+(n-1)(-6)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -20+(n-1)(-6)=-278 \implies n = 44$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{44} \left(- 6 n - 14\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 44(-20-278) }{2}=-6556$ .