Write the sum $LaTeX: \displaystyle 39+42+45 \ldots +111+114$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=42-(39)=3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 39+(n-1)(3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 39+(n-1)(3)=114 \implies n = 26$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{26} \left(3 n + 36\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 26(39+114) }{2}=1989$ .