Write the sum $LaTeX: \displaystyle -5-7-9 \ldots -45-47$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-7-(-5)=-2$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -5+(n-1)(-2)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -5+(n-1)(-2)=-47 \implies n = 22$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{22} \left(- 2 n - 3\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 22(-5-47) }{2}=-572$ .