Write the sum $LaTeX: \displaystyle 42+33+24 \ldots -345-354$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=33-(42)=-9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 42+(n-1)(-9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 42+(n-1)(-9)=-354 \implies n = 45$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{45} \left(51 - 9 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 45(42-354) }{2}=-7020$ .