Write the sum $LaTeX: \displaystyle -8-12-16 \ldots -116-120$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-12-(-8)=-4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -8+(n-1)(-4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -8+(n-1)(-4)=-120 \implies n = 29$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{29} \left(- 4 n - 4\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 29(-8-120) }{2}=-1856$ .