Write the sum $LaTeX: \displaystyle 33+39+45 \ldots +213+219$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=39-(33)=6$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 33+(n-1)(6)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 33+(n-1)(6)=219 \implies n = 32$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{32} \left(6 n + 27\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 32(33+219) }{2}=4032$ .