Write the sum $LaTeX: \displaystyle -6-15-24 \ldots -195-204$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-15-(-6)=-9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -6+(n-1)(-9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -6+(n-1)(-9)=-204 \implies n = 23$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{23} \left(3 - 9 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 23(-6-204) }{2}=-2415$ .