Write the sum $LaTeX: \displaystyle -37-44-51 \ldots -226-233$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-44-(-37)=-7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -37+(n-1)(-7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -37+(n-1)(-7)=-233 \implies n = 29$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{29} \left(- 7 n - 30\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 29(-37-233) }{2}=-3915$ .