Write the sum $LaTeX: \displaystyle -34-41-48 \ldots -279-286$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-41-(-34)=-7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -34+(n-1)(-7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -34+(n-1)(-7)=-286 \implies n = 37$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{37} \left(- 7 n - 27\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 37(-34-286) }{2}=-5920$ .