Write the sum $LaTeX: \displaystyle 23+25+27 \ldots +67+69$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=25-(23)=2$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 23+(n-1)(2)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 23+(n-1)(2)=69 \implies n = 24$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{24} \left(2 n + 21\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 24(23+69) }{2}=1104$ .