Write the sum $LaTeX: \displaystyle -27-18-9 \ldots +270+279$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-18-(-27)=9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -27+(n-1)(9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -27+(n-1)(9)=279 \implies n = 35$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{35} \left(9 n - 36\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 35(-27+279) }{2}=4410$ .