Write the sum $LaTeX: \displaystyle -22-14-6 \ldots +354+362$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-14-(-22)=8$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -22+(n-1)(8)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -22+(n-1)(8)=362 \implies n = 49$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{49} \left(8 n - 30\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 49(-22+362) }{2}=8330$ .