Write the sum $LaTeX: \displaystyle -24-21-18 \ldots +87+90$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-21-(-24)=3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -24+(n-1)(3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -24+(n-1)(3)=90 \implies n = 39$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{39} \left(3 n - 27\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 39(-24+90) }{2}=1287$ .