Write the sum $LaTeX: \displaystyle -3-6-9 \ldots -105-108$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-6-(-3)=-3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -3+(n-1)(-3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -3+(n-1)(-3)=-108 \implies n = 36$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{36} - 3 n$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 36(-3-108) }{2}=-1998$ .