Write the sum $LaTeX: \displaystyle 10+3-4 \ldots -221-228$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=3-(10)=-7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 10+(n-1)(-7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 10+(n-1)(-7)=-228 \implies n = 35$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{35} \left(17 - 7 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 35(10-228) }{2}=-3815$ .