Write the sum $LaTeX: \displaystyle -43-35-27 \ldots +309+317$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-35-(-43)=8$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -43+(n-1)(8)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -43+(n-1)(8)=317 \implies n = 46$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{46} \left(8 n - 51\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 46(-43+317) }{2}=6302$ .