Write the sum $LaTeX: \displaystyle 37+35+33 \ldots -49-51$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=35-(37)=-2$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 37+(n-1)(-2)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 37+(n-1)(-2)=-51 \implies n = 45$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{45} \left(39 - 2 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 45(37-51) }{2}=-315$ .