Write the sum $LaTeX: \displaystyle 20+14+8 \ldots -148-154$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=14-(20)=-6$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 20+(n-1)(-6)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 20+(n-1)(-6)=-154 \implies n = 30$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{30} \left(26 - 6 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 30(20-154) }{2}=-2010$ .