Write the sum $LaTeX: \displaystyle -22-29-36 \ldots -246-253$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-29-(-22)=-7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -22+(n-1)(-7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -22+(n-1)(-7)=-253 \implies n = 34$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{34} \left(- 7 n - 15\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 34(-22-253) }{2}=-4675$ .