Write the sum $LaTeX: \displaystyle 13+9+5 \ldots -171-175$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=9-(13)=-4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 13+(n-1)(-4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 13+(n-1)(-4)=-175 \implies n = 48$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{48} \left(17 - 4 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 48(13-175) }{2}=-3888$ .