Write the sum $LaTeX: \displaystyle 7+2-3 \ldots -208-213$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=2-(7)=-5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 7+(n-1)(-5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 7+(n-1)(-5)=-213 \implies n = 45$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{45} \left(12 - 5 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 45(7-213) }{2}=-4635$ .