Write the sum $LaTeX: \displaystyle -30-31-32 \ldots -67-68$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-31-(-30)=-1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -30+(n-1)(-1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -30+(n-1)(-1)=-68 \implies n = 39$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{39} \left(- n - 29\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 39(-30-68) }{2}=-1911$ .