Write the sum $LaTeX: \displaystyle 42+50+58 \ldots +354+362$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=50-(42)=8$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 42+(n-1)(8)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 42+(n-1)(8)=362 \implies n = 41$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{41} \left(8 n + 34\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 41(42+362) }{2}=8282$ .