Write the sum $LaTeX: \displaystyle 27+21+15 \ldots -219-225$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=21-(27)=-6$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 27+(n-1)(-6)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 27+(n-1)(-6)=-225 \implies n = 43$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{43} \left(33 - 6 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 43(27-225) }{2}=-4257$ .