Write the sum $LaTeX: \displaystyle -46-51-56 \ldots -276-281$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-51-(-46)=-5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -46+(n-1)(-5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -46+(n-1)(-5)=-281 \implies n = 48$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{48} \left(- 5 n - 41\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 48(-46-281) }{2}=-7848$ .