Write the sum $LaTeX: \displaystyle -19-13-7 \ldots +233+239$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-13-(-19)=6$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -19+(n-1)(6)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -19+(n-1)(6)=239 \implies n = 44$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{44} \left(6 n - 25\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 44(-19+239) }{2}=4840$ .