Write the sum $LaTeX: \displaystyle 0-4-8 \ldots -172-176$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-4-(0)=-4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 0+(n-1)(-4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 0+(n-1)(-4)=-176 \implies n = 45$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{45} \left(4 - 4 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 45(0-176) }{2}=-3960$ .