Write the sum $LaTeX: \displaystyle -24-16-8 \ldots +184+192$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-16-(-24)=8$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -24+(n-1)(8)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -24+(n-1)(8)=192 \implies n = 28$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{28} \left(8 n - 32\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 28(-24+192) }{2}=2352$ .