Write the sum $LaTeX: \displaystyle 22+17+12 \ldots -198-203$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=17-(22)=-5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 22+(n-1)(-5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 22+(n-1)(-5)=-203 \implies n = 46$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{46} \left(27 - 5 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 46(22-203) }{2}=-4163$ .