AMC12 2010 12A, Problem 10

The first four terms of an arithmetic sequence are $p$, $9$, $3p-q$, and $3p+q$. What is the $2010^\text{th}$ term of this sequence?

Solution: $3p-q$ and $3p+q$ are consecutive terms, so the common difference is $(3p+q)-(3p-q) = 2q$. \begin{align}p+2q &= 9\\ 9+2q &= 3p-q\\ q&=2\\ p&=5\end{align} The common difference is $4$. The first term is $5$ and the $2010^\text{th}$ term is \[5+4(2009) = \fbox{8041}\]

Answer: 8041