Write the sum $LaTeX: \displaystyle -49-48-47 \ldots -10-9$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-48-(-49)=1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -49+(n-1)(1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -49+(n-1)(1)=-9 \implies n = 41$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{41} \left(n - 50\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 41(-49-9) }{2}=-1189$ .