Write the sum $LaTeX: \displaystyle 43+35+27 \ldots -157-165$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=35-(43)=-8$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 43+(n-1)(-8)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 43+(n-1)(-8)=-165 \implies n = 27$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{27} \left(51 - 8 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 27(43-165) }{2}=-1647$ .