Write the sum $LaTeX: \displaystyle -6+2+10 \ldots +314+322$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=2-(-6)=8$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -6+(n-1)(8)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -6+(n-1)(8)=322 \implies n = 42$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{42} \left(8 n - 14\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 42(-6+322) }{2}=6636$ .