Write the sum $LaTeX: \displaystyle -2-11-20 \ldots -281-290$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-11-(-2)=-9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -2+(n-1)(-9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -2+(n-1)(-9)=-290 \implies n = 33$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{33} \left(7 - 9 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 33(-2-290) }{2}=-4818$ .