Write the sum $LaTeX: \displaystyle -25-32-39 \ldots -284-291$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-32-(-25)=-7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -25+(n-1)(-7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -25+(n-1)(-7)=-291 \implies n = 39$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{39} \left(- 7 n - 18\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 39(-25-291) }{2}=-6162$ .