Write the sum $LaTeX: \displaystyle -47-39-31 \ldots +345+353$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-39-(-47)=8$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -47+(n-1)(8)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -47+(n-1)(8)=353 \implies n = 51$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{51} \left(8 n - 55\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 51(-47+353) }{2}=7803$ .