Write the sum $LaTeX: \displaystyle 24+21+18 \ldots -117-120$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=21-(24)=-3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 24+(n-1)(-3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 24+(n-1)(-3)=-120 \implies n = 49$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{49} \left(27 - 3 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 49(24-120) }{2}=-2352$ .