Write the sum $LaTeX: \displaystyle -45-41-37 \ldots +31+35$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-41-(-45)=4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -45+(n-1)(4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -45+(n-1)(4)=35 \implies n = 21$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{21} \left(4 n - 49\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 21(-45+35) }{2}=-105$ .