Write the sum $LaTeX: \displaystyle -12-20-28 \ldots -236-244$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-20-(-12)=-8$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -12+(n-1)(-8)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -12+(n-1)(-8)=-244 \implies n = 30$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{30} \left(- 8 n - 4\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 30(-12-244) }{2}=-3840$ .