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Difference between revisions of "2009 AMC 12A Problems/Problem 7"

(New page: == Problem == The first three terms of an arithmetic sequence are <math>2x - 3</math>, <math>5x - 11</math>, and <math>3x + 1</math> respectively. The <math>n</math>th term of the sequence...)
 
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== Solution ==
 
== Solution ==
  
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As this is an arithmetic sequence, the difference must be constant: <math>(5x-11) - (2x-3) = (3x+1) - (5x-11)</math>. This solves to <math>x=4</math>. The first three terms then are <math>5</math>, <math>9</math>, and <math>13</math>. In general, the <math>n</math>-th term is <math>1+4n</math>. Solving <math>1+4n=2009</math> we get <math>n=\boxed{502}</math>.
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As this is an arithmetic sequence, the difference must be constant: <math>(5x-11) - (2x-3) = (3x+1) - (5x-11)</math>. This solves to <math>x=4</math>. The first three terms then are <math>5</math>, <math>9</math>, and <math>13</math>. In general, the <math>n</math>th term is <math>1+4n</math>. Solving <math>1+4n=2009</math>, we get <math>n=\boxed{502}</math>.
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== See Also ==
 
== See Also ==
  
 
{{AMC12 box|year=2009|ab=A|num-b=6|num-a=8}}
 
{{AMC12 box|year=2009|ab=A|num-b=6|num-a=8}}
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{{MAA Notice}}

Latest revision as of 09:17, 9 February 2015

Problem

The first three terms of an arithmetic sequence are $2x - 3$, $5x - 11$, and $3x + 1$ respectively. The $n$th term of the sequence is $2009$. What is $n$?

$\textbf{(A)}\ 255 \qquad \textbf{(B)}\ 502 \qquad \textbf{(C)}\ 1004 \qquad \textbf{(D)}\ 1506 \qquad \textbf{(E)}\ 8037$

Solution

As this is an arithmetic sequence, the difference must be constant: $(5x-11) - (2x-3) = (3x+1) - (5x-11)$. This solves to $x=4$. The first three terms then are $5$, $9$, and $13$. In general, the $n$th term is $1+4n$. Solving $1+4n=2009$, we get $n=\boxed{502}$.

See Also

2009 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
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

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