Write the sum $LaTeX: \displaystyle -9-16-23 \ldots -233-240$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-16-(-9)=-7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -9+(n-1)(-7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -9+(n-1)(-7)=-240 \implies n = 34$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{34} \left(- 7 n - 2\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 34(-9-240) }{2}=-4233$ .