Write the sum $LaTeX: \displaystyle -12-21-30 \ldots -354-363$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-21-(-12)=-9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -12+(n-1)(-9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -12+(n-1)(-9)=-363 \implies n = 40$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{40} \left(- 9 n - 3\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 40(-12-363) }{2}=-7500$ .