Write the sum $LaTeX: \displaystyle -21-30-39 \ldots -444-453$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-30-(-21)=-9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -21+(n-1)(-9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -21+(n-1)(-9)=-453 \implies n = 49$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{49} \left(- 9 n - 12\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 49(-21-453) }{2}=-11613$ .