Write the sum $LaTeX: \displaystyle 18+14+10 \ldots -158-162$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=14-(18)=-4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 18+(n-1)(-4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 18+(n-1)(-4)=-162 \implies n = 46$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{46} \left(22 - 4 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 46(18-162) }{2}=-3312$ .