Write the sum $LaTeX: \displaystyle -10-3+4 \ldots +326+333$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-3-(-10)=7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -10+(n-1)(7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -10+(n-1)(7)=333 \implies n = 50$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{50} \left(7 n - 17\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 50(-10+333) }{2}=8075$ .