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

Revision as of 23:16, 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 1: / Line 1: @@
-== Problem 10 ==
+== Problem ==
 The first four terms of an arithmetic sequence are <math>p</math>, <math>9</math>, <math>3p-q</math>, and <math>3p+q</math>. What is the <math>2010^\text{th}</math> term of this sequence?
@@ 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\\
+ &9+2q = 3p-q\\
+ &q=2\\
+ &p=5\end{align*}</cmath>
-<math>p+2q = 9</math>
-<math>9+2q = 3p-q</math>
+The common difference is <math>4</math>. The first term is <math>5</math> and the <math>2010^\text{th}</math> term is
-<math>q=2</math>
-<math>p=5</math>
+<cmath>5+4(2009) = \boxed{8041\ \textbf{(A)}}</cmath>
-The common difference is <math>4</math>. The first term is <math>5</math> and the <math>2010^\text{th}</math> term is
+== See also ==
+{{AMC12 box|year=2010|num-b=9|num-a=11|ab=A}}
-<math>5+4(2009) = \boxed{8041\ \textbf{(A)}}</math>
+[[Category:Introductory Algebra Problems]]