Write the sum $LaTeX: \displaystyle 36+35+34 \ldots +3+2$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=35-(36)=-1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 36+(n-1)(-1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 36+(n-1)(-1)=2 \implies n = 35$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{35} \left(37 - n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 35(36+2) }{2}=665$ .