Write the sum $LaTeX: \displaystyle 14+18+22 \ldots +150+154$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=18-(14)=4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 14+(n-1)(4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 14+(n-1)(4)=154 \implies n = 36$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{36} \left(4 n + 10\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 36(14+154) }{2}=3024$ .