Write the sum $LaTeX: \displaystyle 8+11+14 \ldots +119+122$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=11-(8)=3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 8+(n-1)(3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 8+(n-1)(3)=122 \implies n = 39$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{39} \left(3 n + 5\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 39(8+122) }{2}=2535$ .