Write the sum $LaTeX: \displaystyle -35-41-47 \ldots -149-155$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-41-(-35)=-6$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -35+(n-1)(-6)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -35+(n-1)(-6)=-155 \implies n = 21$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{21} \left(- 6 n - 29\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 21(-35-155) }{2}=-1995$ .