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Difference between revisions of "2001 AMC 12 Problems/Problem 25"

(New page: == Problem == Consider sequences of positive real numbers of the form <math>x, 2000, y, \dots</math> in which every term after the first is 1 less than the product of its two immediate ne...)
 
(→‎Remark)
 
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== Problem ==
 
== Problem ==
  
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Consider sequences of positive real numbers of the form <math>x, 2000, y, \dots</math> in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of <math>x</math> does the term 2001 appear somewhere in the sequence?
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Consider sequences of positive real numbers of the form <math>x, 2000, y, \dots</math> in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of <math>x</math> does the term <math>2001</math> appear somewhere in the sequence?
  
 
<math>
 
<math>
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== Solution ==
 
== Solution ==
  
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It never hurts to compute a few terms of the sequence in order to get a feel how it looks like. In our case, the definition is that <math>\forall n>1:~ a_n = a_{n-1}a_{n+1} - 1</math>. This can be rewritten as <math>a_{n+1} = \frac{a_n +1}{a_{n-1}}</math>. We have <math>a_1=x</math> and <math>a_2=2000</math>, and we compute:
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It never hurts to compute a few terms of the sequence in order to get a feel how it looks like. In our case, the definition is that <math>\forall</math> (for all) <math> n>1:~ a_n = a_{n-1}a_{n+1} - 1</math>. This can be rewritten as <math>a_{n+1} = \frac{a_n +1}{a_{n-1}}</math>. We have <math>a_1=x</math> and <math>a_2=2000</math>, and we compute:
  
 
<cmath>
 
<cmath>
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& = \frac{a_4+1}{a_3}
 
& = \frac{a_4+1}{a_3}
 
= \frac{ \frac{2001 + x}{2000x} + 1 }{ \frac{2001}x }
 
= \frac{ \frac{2001 + x}{2000x} + 1 }{ \frac{2001}x }
−
= \frac{ \frac{2001 + 2001x} }{ 2000\cdot 2001 }
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= \frac{ \frac{2001 + 2001x}{2000x} }{ \frac{2001}x }
 
= \frac{1+x}{2000}
 
= \frac{1+x}{2000}
 
\\
 
\\
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No two values of <math>x</math> we just computed are equal, and therefore there are <math>\boxed{4}</math> different values of <math>x</math> for which the sequence contains the value <math>2001</math>.
 
No two values of <math>x</math> we just computed are equal, and therefore there are <math>\boxed{4}</math> different values of <math>x</math> for which the sequence contains the value <math>2001</math>.
  
 +
==Remark==
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 +
In general, notice that the sequence defined by <math>a_1=a, a_2=b</math>, and <math>a_{n+1} = \frac{a_n +1}{a_{n-1}}</math> for <math>n\ge 2</math> is periodic.
 +
 +
~tsun26
  
 
== See Also ==
 
== See Also ==
  
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{{AMC12 box|year=2001|num-b=|num-a=}}
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{{AMC12 box|year=2001|num-b=24|after=Last question}}
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{{MAA Notice}}

Latest revision as of 02:10, 5 November 2024

Problem

Consider sequences of positive real numbers of the form $x, 2000, y, \dots$ in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of $x$ does the term $2001$ appear somewhere in the sequence?

$\text{(A) }1 \qquad \text{(B) }2 \qquad \text{(C) }3 \qquad \text{(D) }4 \qquad \text{(E) more than }4$

Solution

It never hurts to compute a few terms of the sequence in order to get a feel how it looks like. In our case, the definition is that $\forall$ (for all) $n>1:~ a_n = a_{n-1}a_{n+1} - 1$. This can be rewritten as $a_{n+1} = \frac{a_n +1}{a_{n-1}}$. We have $a_1=x$ and $a_2=2000$, and we compute:

\begin{align*} a_3 & = \frac{a_2+1}{a_1} = \frac{2001}x \\ a_4 & = \frac{a_3+1}{a_2} = \frac{ \dfrac{2001}x + 1 }{ 2000 } = \frac{2001 + x}{2000x} \\ a_5 & = \frac{a_4+1}{a_3} = \frac{ \frac{2001 + x}{2000x} + 1 }{ \frac{2001}x } = \frac{ \frac{2001 + 2001x}{2000x} }{ \frac{2001}x } = \frac{1+x}{2000} \\ a_6 & = \frac{a_5+1}{a_4} = \frac{ \frac{1+x}{2000} + 1 }{ \frac{2001 + x}{2000x} } = \frac{ \frac{2001+x}{2000} }{ \frac{2001 + x}{2000x} } = x \\ a_7 & = \frac{a_6+1}{a_5} = \frac{ x+1 }{ \frac{1+x}{2000} } = 2000 \end{align*}

At this point we see that the sequence will become periodic: we have $a_6=a_1$, $a_7=a_2$, and each subsequent term is uniquely determined by the previous two.

Hence if $2001$ appears, it has to be one of $a_1$ to $a_5$. As $a_2=2000$, we only have four possibilities left. Clearly $a_1=2001$ for $x=2001$, and $a_3=2001$ for $x=1$. The equation $a_4=2001$ solves to $x = \frac{2001}{2000\cdot 2001 - 1}$, and the equation $a_5=2001$ to $x=2000\cdot 2001 - 1$.

No two values of $x$ we just computed are equal, and therefore there are $\boxed{4}$ different values of $x$ for which the sequence contains the value $2001$.

Remark

In general, notice that the sequence defined by $a_1=a, a_2=b$, and $a_{n+1} = \frac{a_n +1}{a_{n-1}}$ for $n\ge 2$ is periodic.

~tsun26

See Also

2001 AMC 12 (Problems • Answer Key • Resources)
Preceded by
Problem 24 		Followed by
Last question
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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