Write the sum $LaTeX: \displaystyle -12-17-22 \ldots -222-227$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-17-(-12)=-5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -12+(n-1)(-5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -12+(n-1)(-5)=-227 \implies n = 44$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{44} \left(- 5 n - 7\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 44(-12-227) }{2}=-5258$ .