Write the sum $LaTeX: \displaystyle 22+27+32 \ldots +207+212$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=27-(22)=5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 22+(n-1)(5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 22+(n-1)(5)=212 \implies n = 39$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{39} \left(5 n + 17\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 39(22+212) }{2}=4563$ .