Write the sum $LaTeX: \displaystyle 31+24+17 \ldots -221-228$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=24-(31)=-7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 31+(n-1)(-7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 31+(n-1)(-7)=-228 \implies n = 38$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{38} \left(38 - 7 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 38(31-228) }{2}=-3743$ .