Write the sum $LaTeX: \displaystyle -16-13-10 \ldots +131+134$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-13-(-16)=3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -16+(n-1)(3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -16+(n-1)(3)=134 \implies n = 51$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{51} \left(3 n - 19\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 51(-16+134) }{2}=3009$ .