Write the sum $LaTeX: \displaystyle -45-52-59 \ldots -318-325$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-52-(-45)=-7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -45+(n-1)(-7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -45+(n-1)(-7)=-325 \implies n = 41$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{41} \left(- 7 n - 38\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 41(-45-325) }{2}=-7585$ .