Write the sum $LaTeX: \displaystyle -48-55-62 \ldots -223-230$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-55-(-48)=-7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -48+(n-1)(-7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -48+(n-1)(-7)=-230 \implies n = 27$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{27} \left(- 7 n - 41\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 27(-48-230) }{2}=-3753$ .