Write the sum $LaTeX: \displaystyle -7-13-19 \ldots -241-247$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-13-(-7)=-6$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -7+(n-1)(-6)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -7+(n-1)(-6)=-247 \implies n = 41$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{41} \left(- 6 n - 1\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 41(-7-247) }{2}=-5207$ .