Write the sum $LaTeX: \displaystyle -18-25-32 \ldots -165-172$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-25-(-18)=-7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -18+(n-1)(-7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -18+(n-1)(-7)=-172 \implies n = 23$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{23} \left(- 7 n - 11\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 23(-18-172) }{2}=-2185$ .