Write the sum $LaTeX: \displaystyle -9-3+3 \ldots +105+111$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-3-(-9)=6$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -9+(n-1)(6)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -9+(n-1)(6)=111 \implies n = 21$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{21} \left(6 n - 15\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 21(-9+111) }{2}=1071$ .