Write the sum $LaTeX: \displaystyle -7-14-21 \ldots -245-252$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-14-(-7)=-7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -7+(n-1)(-7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -7+(n-1)(-7)=-252 \implies n = 36$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{36} - 7 n$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 36(-7-252) }{2}=-4662$ .