Write the sum $LaTeX: \displaystyle -41-34-27 \ldots +92+99$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-34-(-41)=7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -41+(n-1)(7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -41+(n-1)(7)=99 \implies n = 21$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{21} \left(7 n - 48\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 21(-41+99) }{2}=609$ .