Write the sum $LaTeX: \displaystyle 47+41+35 \ldots -97-103$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=41-(47)=-6$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 47+(n-1)(-6)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 47+(n-1)(-6)=-103 \implies n = 26$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{26} \left(53 - 6 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 26(47-103) }{2}=-728$ .