Write the sum $LaTeX: \displaystyle -26-22-18 \ldots +118+122$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-22-(-26)=4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -26+(n-1)(4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -26+(n-1)(4)=122 \implies n = 38$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{38} \left(4 n - 30\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 38(-26+122) }{2}=1824$ .