Write the sum $LaTeX: \displaystyle 22+24+26 \ldots +72+74$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=24-(22)=2$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 22+(n-1)(2)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 22+(n-1)(2)=74 \implies n = 27$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{27} \left(2 n + 20\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 27(22+74) }{2}=1296$ .