Write the sum $LaTeX: \displaystyle 0-6-12 \ldots -222-228$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-6-(0)=-6$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 0+(n-1)(-6)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 0+(n-1)(-6)=-228 \implies n = 39$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{39} \left(6 - 6 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 39(0-228) }{2}=-4446$ .