Write the sum $LaTeX: \displaystyle 23+22+21 \ldots -9-10$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=22-(23)=-1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 23+(n-1)(-1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 23+(n-1)(-1)=-10 \implies n = 34$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{34} \left(24 - n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 34(23-10) }{2}=221$ .