Write the sum $LaTeX: \displaystyle 4+1-2 \ldots -104-107$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=1-(4)=-3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 4+(n-1)(-3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 4+(n-1)(-3)=-107 \implies n = 38$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{38} \left(7 - 3 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 38(4-107) }{2}=-1957$ .