Write the sum $LaTeX: \displaystyle 15+14+13 \ldots -11-12$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=14-(15)=-1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 15+(n-1)(-1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 15+(n-1)(-1)=-12 \implies n = 28$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{28} \left(16 - n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 28(15-12) }{2}=42$ .