Write the sum $LaTeX: \displaystyle -27-29-31 \ldots -65-67$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-29-(-27)=-2$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -27+(n-1)(-2)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -27+(n-1)(-2)=-67 \implies n = 21$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{21} \left(- 2 n - 25\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 21(-27-67) }{2}=-987$ .