Write the sum $LaTeX: \displaystyle 30+24+18 \ldots -204-210$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=24-(30)=-6$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 30+(n-1)(-6)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 30+(n-1)(-6)=-210 \implies n = 41$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{41} \left(36 - 6 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 41(30-210) }{2}=-3690$ .