Write the sum $LaTeX: \displaystyle 17+24+31 \ldots +206+213$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=24-(17)=7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 17+(n-1)(7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 17+(n-1)(7)=213 \implies n = 29$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{29} \left(7 n + 10\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 29(17+213) }{2}=3335$ .