Write the sum $LaTeX: \displaystyle 25+32+39 \ldots +312+319$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=32-(25)=7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 25+(n-1)(7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 25+(n-1)(7)=319 \implies n = 43$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{43} \left(7 n + 18\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 43(25+319) }{2}=7396$ .