Write the sum $LaTeX: \displaystyle -14-13-12 \ldots +34+35$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-13-(-14)=1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -14+(n-1)(1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -14+(n-1)(1)=35 \implies n = 50$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{50} \left(n - 15\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 50(-14+35) }{2}=525$ .