Write the sum $LaTeX: \displaystyle 42+47+52 \ldots +232+237$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=47-(42)=5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 42+(n-1)(5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 42+(n-1)(5)=237 \implies n = 40$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{40} \left(5 n + 37\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 40(42+237) }{2}=5580$ .