Write the sum $LaTeX: \displaystyle -25-20-15 \ldots +220+225$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-20-(-25)=5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -25+(n-1)(5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -25+(n-1)(5)=225 \implies n = 51$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{51} \left(5 n - 30\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 51(-25+225) }{2}=5100$ .