Write the sum $LaTeX: \displaystyle -14-20-26 \ldots -254-260$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-20-(-14)=-6$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -14+(n-1)(-6)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -14+(n-1)(-6)=-260 \implies n = 42$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{42} \left(- 6 n - 8\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 42(-14-260) }{2}=-5754$ .