Write the sum $LaTeX: \displaystyle -38-42-46 \ldots -182-186$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-42-(-38)=-4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -38+(n-1)(-4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -38+(n-1)(-4)=-186 \implies n = 38$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{38} \left(- 4 n - 34\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 38(-38-186) }{2}=-4256$ .