Write the sum $LaTeX: \displaystyle 27+19+11 \ldots -349-357$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=19-(27)=-8$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 27+(n-1)(-8)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 27+(n-1)(-8)=-357 \implies n = 49$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{49} \left(35 - 8 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 49(27-357) }{2}=-8085$ .