2010 AMC 12A Problems/Problem 10

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?

$\textbf{(A)}\ 8041 \qquad \textbf{(B)}\ 8043 \qquad \textbf{(C)}\ 8045 \qquad \textbf{(D)}\ 8047 \qquad \textbf{(E)}\ 8049$

$3p-q$ and $3p+q$ are consecutive terms, so the common difference is $(3p+q)-(3p-q) = 2q$ .

$p+2q = 9$

$9+2q = 3p-q$

$q=2$

$p=5$

The common difference is $4$ . The first term is $5$ and the $2010^\text{th}$ term is

$5+4(2009) = \boxed{8041\ \textbf{(A)}}$