Write the sum $LaTeX: \displaystyle -30-38-46 \ldots -230-238$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-38-(-30)=-8$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -30+(n-1)(-8)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -30+(n-1)(-8)=-238 \implies n = 27$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{27} \left(- 8 n - 22\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 27(-30-238) }{2}=-3618$ .