Write the sum $LaTeX: \displaystyle 38+46+54 \ldots +286+294$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=46-(38)=8$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 38+(n-1)(8)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 38+(n-1)(8)=294 \implies n = 33$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{33} \left(8 n + 30\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 33(38+294) }{2}=5478$ .