Write the sum $LaTeX: \displaystyle -4-2+0 \ldots +82+84$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-2-(-4)=2$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -4+(n-1)(2)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -4+(n-1)(2)=84 \implies n = 45$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{45} \left(2 n - 6\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 45(-4+84) }{2}=1800$ .