Write the sum $LaTeX: \displaystyle -37-36-35 \ldots -9-8$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-36-(-37)=1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -37+(n-1)(1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -37+(n-1)(1)=-8 \implies n = 30$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{30} \left(n - 38\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 30(-37-8) }{2}=-675$ .