Write the sum $LaTeX: \displaystyle 5+0-5 \ldots -120-125$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=0-(5)=-5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 5+(n-1)(-5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 5+(n-1)(-5)=-125 \implies n = 27$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{27} \left(10 - 5 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 27(5-125) }{2}=-1620$ .