Write the sum $LaTeX: \displaystyle 3-6-15 \ldots -168-177$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-6-(3)=-9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 3+(n-1)(-9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 3+(n-1)(-9)=-177 \implies n = 21$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{21} \left(12 - 9 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 21(3-177) }{2}=-1827$ .