Write the sum $LaTeX: \displaystyle 31+30+29 \ldots -6-7$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=30-(31)=-1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 31+(n-1)(-1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 31+(n-1)(-1)=-7 \implies n = 39$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{39} \left(32 - n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 39(31-7) }{2}=468$ .