Write the sum $LaTeX: \displaystyle -37-41-45 \ldots -169-173$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-41-(-37)=-4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -37+(n-1)(-4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -37+(n-1)(-4)=-173 \implies n = 35$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{35} \left(- 4 n - 33\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 35(-37-173) }{2}=-3675$ .