Write the sum $LaTeX: \displaystyle 5+10+15 \ldots +250+255$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=10-(5)=5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 5+(n-1)(5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 5+(n-1)(5)=255 \implies n = 51$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{51} 5 n$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 51(5+255) }{2}=6630$ .