Write the sum $LaTeX: \displaystyle -29-20-11 \ldots +232+241$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-20-(-29)=9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -29+(n-1)(9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -29+(n-1)(9)=241 \implies n = 31$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{31} \left(9 n - 38\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 31(-29+241) }{2}=3286$ .