Write the sum $LaTeX: \displaystyle 40+39+38 \ldots +9+8$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=39-(40)=-1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 40+(n-1)(-1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 40+(n-1)(-1)=8 \implies n = 33$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{33} \left(41 - n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 33(40+8) }{2}=792$ .