Write the sum $LaTeX: \displaystyle -19-22-25 \ldots -145-148$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-22-(-19)=-3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -19+(n-1)(-3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -19+(n-1)(-3)=-148 \implies n = 44$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{44} \left(- 3 n - 16\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 44(-19-148) }{2}=-3674$ .