Write the sum $LaTeX: \displaystyle 18+11+4 \ldots -206-213$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=11-(18)=-7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 18+(n-1)(-7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 18+(n-1)(-7)=-213 \implies n = 34$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{34} \left(25 - 7 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 34(18-213) }{2}=-3315$ .