Write the sum $LaTeX: \displaystyle 10+7+4 \ldots -62-65$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=7-(10)=-3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 10+(n-1)(-3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 10+(n-1)(-3)=-65 \implies n = 26$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{26} \left(13 - 3 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 26(10-65) }{2}=-715$ .