Write the sum $LaTeX: \displaystyle -6-2+2 \ldots +122+126$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-2-(-6)=4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -6+(n-1)(4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -6+(n-1)(4)=126 \implies n = 34$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{34} \left(4 n - 10\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 34(-6+126) }{2}=2040$ .