Write the sum $LaTeX: \displaystyle 3+5+7 \ldots +51+53$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=5-(3)=2$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 3+(n-1)(2)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 3+(n-1)(2)=53 \implies n = 26$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{26} \left(2 n + 1\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 26(3+53) }{2}=728$ .