Write the sum $LaTeX: \displaystyle -4-8-12 \ldots -164-168$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-8-(-4)=-4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -4+(n-1)(-4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -4+(n-1)(-4)=-168 \implies n = 42$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{42} - 4 n$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 42(-4-168) }{2}=-3612$ .