Write the sum $LaTeX: \displaystyle -5-9-13 \ldots -153-157$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-9-(-5)=-4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -5+(n-1)(-4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -5+(n-1)(-4)=-157 \implies n = 39$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{39} \left(- 4 n - 1\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 39(-5-157) }{2}=-3159$ .