Write the sum $LaTeX: \displaystyle -45-44-43 \ldots -21-20$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-44-(-45)=1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -45+(n-1)(1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -45+(n-1)(1)=-20 \implies n = 26$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{26} \left(n - 46\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 26(-45-20) }{2}=-845$ .