Difference between revisions of "2010 AMC 12A Problems/Problem 10"

Revision as of 22:27, 25 February 2010

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 $(3p+q)-(3p-q) = 2q$ .

$\begin{align*}p+2q &= 9\\ 9+2q &= 3p-q\\ q&=2\\ p&=5\end{align*}$

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

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

2010 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

@@ Line 7: / Line 7: @@
 <math>3p-q</math> and <math>3p+q</math> are consecutive terms, so the common difference is <math>(3p+q)-(3p-q) = 2q</math>.
-<cmath>\begin{align*}&p+2q = 9\\
+<cmath>\begin{align*}p+2q &= 9\\
- &9+2q = 3p-q\\
++2q &= 3p-q\\
- &q=2\\
+q&=2\\
- &p=5\end{align*}</cmath>
+p&=5\end{align*}</cmath>
 The common difference is <math>4</math>. The first term is <math>5</math> and the <math>2010^\text{th}</math> term is