Write the sum $LaTeX: \displaystyle 36+42+48 \ldots +168+174$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=42-(36)=6$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 36+(n-1)(6)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 36+(n-1)(6)=174 \implies n = 24$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{24} \left(6 n + 30\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 24(36+174) }{2}=2520$ .