Write the sum $LaTeX: \displaystyle 4-1-6 \ldots -111-116$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-1-(4)=-5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 4+(n-1)(-5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 4+(n-1)(-5)=-116 \implies n = 25$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{25} \left(9 - 5 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 25(4-116) }{2}=-1400$ .