Write the sum $LaTeX: \displaystyle 33+32+31 \ldots +13+12$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=32-(33)=-1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 33+(n-1)(-1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 33+(n-1)(-1)=12 \implies n = 22$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{22} \left(34 - n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 22(33+12) }{2}=495$ .