Write the sum $LaTeX: \displaystyle -42-41-40 \ldots -3-2$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-41-(-42)=1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -42+(n-1)(1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -42+(n-1)(1)=-2 \implies n = 41$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{41} \left(n - 43\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 41(-42-2) }{2}=-902$ .