Write the sum $LaTeX: \displaystyle -45-43-41 \ldots +9+11$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-43-(-45)=2$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -45+(n-1)(2)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -45+(n-1)(2)=11 \implies n = 29$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{29} \left(2 n - 47\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 29(-45+11) }{2}=-493$ .