Write the sum $LaTeX: \displaystyle 6+15+24 \ldots +222+231$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=15-(6)=9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 6+(n-1)(9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 6+(n-1)(9)=231 \implies n = 26$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{26} \left(9 n - 3\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 26(6+231) }{2}=3081$ .