Write the sum $LaTeX: \displaystyle -39-42-45 \ldots -102-105$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-42-(-39)=-3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -39+(n-1)(-3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -39+(n-1)(-3)=-105 \implies n = 23$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{23} \left(- 3 n - 36\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 23(-39-105) }{2}=-1656$ .