Write the sum $LaTeX: \displaystyle -48-52-56 \ldots -232-236$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-52-(-48)=-4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -48+(n-1)(-4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -48+(n-1)(-4)=-236 \implies n = 48$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{48} \left(- 4 n - 44\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 48(-48-236) }{2}=-6816$ .