Write the sum $LaTeX: \displaystyle -34-37-40 \ldots -178-181$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-37-(-34)=-3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -34+(n-1)(-3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -34+(n-1)(-3)=-181 \implies n = 50$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{50} \left(- 3 n - 31\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 50(-34-181) }{2}=-5375$ .