Write the sum $LaTeX: \displaystyle 36+33+30 \ldots -60-63$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=33-(36)=-3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 36+(n-1)(-3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 36+(n-1)(-3)=-63 \implies n = 34$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{34} \left(39 - 3 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 34(36-63) }{2}=-459$ .