Write the sum $LaTeX: \displaystyle -14-19-24 \ldots -134-139$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-19-(-14)=-5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -14+(n-1)(-5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -14+(n-1)(-5)=-139 \implies n = 26$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{26} \left(- 5 n - 9\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 26(-14-139) }{2}=-1989$ .