Write the sum $LaTeX: \displaystyle -35-32-29 \ldots +103+106$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-32-(-35)=3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -35+(n-1)(3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -35+(n-1)(3)=106 \implies n = 48$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{48} \left(3 n - 38\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 48(-35+106) }{2}=1704$ .