Write the sum $LaTeX: \displaystyle -1-3-5 \ldots -97-99$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-3-(-1)=-2$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -1+(n-1)(-2)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -1+(n-1)(-2)=-99 \implies n = 50$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{50} \left(1 - 2 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 50(-1-99) }{2}=-2500$ .