Write the sum $LaTeX: \displaystyle 18+15+12 \ldots -111-114$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=15-(18)=-3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 18+(n-1)(-3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 18+(n-1)(-3)=-114 \implies n = 45$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{45} \left(21 - 3 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 45(18-114) }{2}=-2160$ .