Write the sum $LaTeX: \displaystyle -34-32-30 \ldots +32+34$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-32-(-34)=2$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -34+(n-1)(2)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -34+(n-1)(2)=34 \implies n = 35$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{35} \left(2 n - 36\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 35(-34+34) }{2}=0$ .