Write the sum $LaTeX: \displaystyle 47+40+33 \ldots -142-149$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=40-(47)=-7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 47+(n-1)(-7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 47+(n-1)(-7)=-149 \implies n = 29$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{29} \left(54 - 7 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 29(47-149) }{2}=-1479$ .