Write the sum $LaTeX: \displaystyle -7-8-9 \ldots -31-32$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-8-(-7)=-1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -7+(n-1)(-1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -7+(n-1)(-1)=-32 \implies n = 26$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{26} \left(- n - 6\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 26(-7-32) }{2}=-507$ .