Write the sum $LaTeX: \displaystyle 21+15+9 \ldots -147-153$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=15-(21)=-6$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 21+(n-1)(-6)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 21+(n-1)(-6)=-153 \implies n = 30$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{30} \left(27 - 6 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 30(21-153) }{2}=-1980$ .