Write the sum $LaTeX: \displaystyle -39-47-55 \ldots -399-407$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-47-(-39)=-8$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -39+(n-1)(-8)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -39+(n-1)(-8)=-407 \implies n = 47$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{47} \left(- 8 n - 31\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 47(-39-407) }{2}=-10481$ .