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Difference between revisions of "2019 AMC 10B Problems/Problem 24"

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Define a sequence recursively by <math>x_0=5</math> and <cmath>x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}</cmath> for all nonnegative integers <math>n.</math> Let <math>m</math> be the least positive integer such that
 
Define a sequence recursively by <math>x_0=5</math> and <cmath>x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}</cmath> for all nonnegative integers <math>n.</math> Let <math>m</math> be the least positive integer such that
<cmath>x_m\leq 4+\frac{1}{2^{20}}.</cmath>In which of the following intervals does <math>m</math> lie?
+
 
 +
<cmath>x_m\leq 4+\frac{1}{2^{20}}.</cmath>
 +
 
 +
In which of the following intervals does <math>m</math> lie?
  
 
<math>\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty)</math>
 
<math>\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty)</math>
  
 
== Solution 1 ==
 
== Solution 1 ==
We first prove that <math>x_n > 4</math> for all <math>n \ge 0</math> by induction from
+
We first prove that <math>x_n > 4</math> for all <math>n \ge 0</math>, by induction. Observe that
 
<cmath>
 
<cmath>
 
x_{n+1} - 4 = \frac{x_n^2 + 5x_n + 4 - 4(x_n+6)}{x_n+6} = \frac{(x_n - 4)(x_n+5)}{x_n+6}
 
x_{n+1} - 4 = \frac{x_n^2 + 5x_n + 4 - 4(x_n+6)}{x_n+6} = \frac{(x_n - 4)(x_n+5)}{x_n+6}
 
</cmath>
 
</cmath>
and then prove <math>x_n</math>'s are decreasing by
+
so (since <math>x_n</math> is clearly positive for all <math>n</math>, from the initial definition), <math>x_{n+1} > 4</math> if and only if <math>x_{n} > 4</math>.
 +
 
 +
We similarly prove that <math>x_n</math> is decreasing, since
 
<cmath>
 
<cmath>
 
x_{n+1} - x_n = \frac{x_n^2 + 5x_n + 4 - x_n(x_n+6)}{x_n+6} = \frac{4-x_n}{x_n+6} < 0
 
x_{n+1} - x_n = \frac{x_n^2 + 5x_n + 4 - x_n(x_n+6)}{x_n+6} = \frac{4-x_n}{x_n+6} < 0
 
</cmath>
 
</cmath>
Now we need to estimate the value of <math>x_{n+1}-4</math> by
+
 
 +
Now we need to estimate the value of <math>x_{n+1}-4</math>, which we can do using the rearranged equation
 
<cmath>
 
<cmath>
 
x_{n+1} - 4 = (x_n-4)\cdot\frac{x_n + 5}{x_n+6}  
 
x_{n+1} - 4 = (x_n-4)\cdot\frac{x_n + 5}{x_n+6}  
 
</cmath>
 
</cmath>
since <math>x_n</math>'s are decreasing, <math>\frac{x_n + 5}{x_n+6}</math> are also decreasing, so we have
+
Since <math>x_n</math> is decreasing, <math>\frac{x_n + 5}{x_n+6}</math> is clearly also decreasing, so we have
 
<cmath>
 
<cmath>
 
\frac{9}{10} < \frac{x_n + 5}{x_n+6} \le \frac{10}{11}
 
\frac{9}{10} < \frac{x_n + 5}{x_n+6} \le \frac{10}{11}
Line 29: Line 35:
 
\frac{9}{10}(x_n-4) < x_{n+1} - 4 \le \frac{10}{11}(x_n-4)
 
\frac{9}{10}(x_n-4) < x_{n+1} - 4 \le \frac{10}{11}(x_n-4)
 
</cmath>
 
</cmath>
which leads to
+
 
 +
This becomes
 
<cmath>
 
<cmath>
(\frac{9}{10})^n = (\frac{9}{10})^n (x_0-4) < x_{n} - 4 \le (\frac{10}{11})^n (x_0-4) = (\frac{10}{11})^n
+
\left(\frac{9}{10}\right)^n = \left(\frac{9}{10}\right)^n \left(x_0-4\right) < x_{n} - 4 \le \left(\frac{10}{11}\right)^n \left(x_0-4\right) = \left(\frac{10}{11}\right)^n
 
</cmath>
 
</cmath>
The problem requires us to find the value of <math>n</math> such that
+
The problem thus reduces to finding the least value of <math>n</math> such that
 
<cmath>
 
<cmath>
(\frac{9}{10})^n < x_{n} - 4 \le \frac{1}{2^{20}} \text{  and  }  
+
\left(\frac{9}{10}\right)^n < x_{n} - 4 \le \frac{1}{2^{20}} \text{  and  }  
(\frac{10}{11})^{n-1} > x_{n-1} - 4 > \frac{1}{2^{20}}
+
\left(\frac{10}{11}\right)^{n-1} > x_{n-1} - 4 > \frac{1}{2^{20}}
 
</cmath>
 
</cmath>
using natural logarithm, we need
+
 
<math>n \ln \frac{9}{10} < -20 \ln 2</math> and <math>(n-1)\ln \frac{10}{11} > -20 \ln 2</math>, or
+
Taking logarithms, we get
 +
<math>n \ln \frac{9}{10} < -20 \ln 2</math> and <math>(n-1)\ln \frac{10}{11} > -20 \ln 2</math>, i.e.
  
 
<cmath>
 
<cmath>
Line 45: Line 53:
 
</cmath>
 
</cmath>
  
As estimations, <math>\ln\frac{10}{9} \approx 1/9</math> and <math>\ln\frac{11}{10} \approx 1/10</math>, <math>\ln 2\approx 0.7</math>
+
As approximations, we can use <math>\ln\frac{10}{9} \approx \frac{1}{9}</math>, <math>\ln\frac{11}{10} \approx \frac{1}{10}</math>, and <math>\ln 2\approx 0.7</math>. These allow us to estimate that
we can estimate that
 
 
<cmath>
 
<cmath>
 
             126 < n < 141
 
             126 < n < 141
 
</cmath>
 
</cmath>
Choose <math>\boxed{C}</math>
+
which gives the answer as <math>\boxed{\textbf{(C) } [81,242]}</math>.
 +
 
 +
==Solution 2==
 +
 
 +
The condition where <math>x_m\leq 4+\frac{1}{2^{20}}</math> gives the motivation to make a substitution to change the equilibrium from <math>4</math> to <math>0</math>. We can substitute <math>x_n = y_n + 4</math> to achieve that. Now, we need to find the smallest value of <math>m</math> such that <math>y_m\leq \frac{1}{2^{20}}</math> given that <math>y_0 = 1</math>.
 +
 
 +
 
 +
 
 +
Factoring the recursion <math>x_{n+1} = \frac{x_n^2 + 5x_n+4}{x_n + 6}</math>, we get:
 +
 
 +
<math>x_{n+1}=\dfrac{(x_n + 4)(x_n + 1)}{x_n + 6} \Rightarrow y_{n+1}+4=\dfrac{(y_n+8)(y_n+5)}{y_n+10}</math>
 +
 
 +
<math>y_{n+1}+4=\dfrac{y_n^2+13y_n+40}{y_n+10} = \dfrac{y_n^2+9y_n +(4y_n+40)}{y_n+10}</math>
 +
 
 +
<math>y_{n+1}+4=\dfrac{y_n^2+9y_n}{y_n+10} + 4</math>
 +
 
 +
<math>y_{n+1}=\dfrac{y_n^2+9y_n}{y_n+10}</math>.
 +
 
 +
 
 +
 
 +
Using wishful thinking, we can simplify the recursion as follows:
  
== Solution 2 ==
+
<math>y_{n+1} = \frac{y_n^2 + 9y_n + y_n - y_n}{y_n + 10}</math>
  
Making the reasonable assumption that <math>x_n</math> approaches <math>4</math>, we can translate <math>x</math> down by <math>4</math> to obtain a more simple sequence <math>a_n=x_n-4</math> that should approach  <math>0</math>.
+
<math>y_{n+1} = \frac{y_n(y_n + 10) - y_n}{y_n + 10}</math>
  
 +
<math>y_{n+1} = y_n - \frac{y_n}{y_n + 10}</math>
  
Substitution of <math>(a_{n}+4)</math> for <math>(x_{n})</math> and <math>(a_{n+1}+4)</math> for <math>(x_{n+1})</math> in the definition of <math>x_{n+1}</math> leads to
+
<math>y_{n+1} = y_n\left(1 - \frac{1}{y_n + 10}\right)</math>.
  
  
<math>a_{n+1}+4 = \frac{(a_{n} + 8)(a_{n}+5)}{a_{n}+10} =  \frac{a_{n}^2 + 13a_{n}+40}{a_{n}+10} \implies a_{n+1} = a_{n}(\frac{a_{n}+9}{a_{n}+10}) \implies \frac{a_{n+1}}{a_{n}} = \frac{a_{n}+9}{a_{n}+10}</math>
+
The recursion looks like a geometric sequence with the ratio changing slightly after each term. Notice from the recursion that the <math>y_n</math> sequence is strictly decreasing, so all the terms after <math>y_0</math> will be less than 1. Also, notice that all the terms in sequence will be positive. Both of these can be proven by induction.
  
  
The ratio of consecutive terms is thus always positive and less than 1 (because <math>a_0</math> is positive). This means that the largest possible value for <math>a_n</math> is 1 and that no value of <math>a_n</math> can be less than or equal to 0.
+
With both of those observations in mind, <math>\frac{9}{10} < 1 - \frac{1}{y_n + 10} \leq \frac{10}{11}</math>. Combining this with the fact that the recursion resembles a geometric sequence, we conclude that <math>\left(\frac{9}{10}\right)^n < y_n \leq \left(\frac{10}{11}\right)^n.</math>
  
 +
<math>\frac{9}{10}</math> is approximately equal to <math>\frac{10}{11}</math> and the ranges that the answer choices give us are generous, so we should use either <math>\frac{9}{10}</math> or <math>\frac{10}{11}</math> to find a rough estimate for <math>m</math>.
  
Plugging the extrema of <math>a_n</math> back into the ratio shows that <math>\frac{9}{10}<\frac{a_{n+1}}{a_{n}}\leq\frac{10}{11}</math> for all <math>n</math>.
 
  
 +
Since <math>\dfrac{1}{2}=0.5</math>, that means <math>\frac{1}{\sqrt{2}}=2^{-\frac{1}{2}} \approx 0.7</math>. Additionally, <math>\left(\frac{9}{10}\right)^3=0.729</math>
  
For <math>(n>0)</math>, we can bound <math>a_{n}</math> by applying this rule recursively  :  <math>(\frac{9}{10})^n < a_{n} \leq (\frac{10}{11})^n</math> 
 
  
 +
Therefore, we can estimate that <math>2^{-\frac{1}{2}} < y_3</math>.
  
 +
Raising both sides to the 40th power, we get <math>2^{-20} < (y_3)^{40}</math>
  
Therefore, <math>a_{n}</math> is always less than <math>(\frac{1}{2^{20}})</math> when <math>(\frac{10}{11})^n<\frac{1}{2^{20}}\implies n>20log_{\frac{11}{10}}{2}</math>
+
But <math>y_3 = (y_0)^3</math>, so <math>2^{-20} < (y_0)^{120}</math> and therefore, <math>2^{-20} < y_{120}</math>.
  
and <math>a_{n}</math> is never less than <math>(\frac{1}{2^{20}})</math> when <math>(\frac{9}{10})^n>\frac{1}{2^{20}}\implies n<20log_{\frac{10}{9}}{2}</math>
+
This tells us that <math>m</math> is somewhere around 120, so our answer is <math>\boxed{\textbf{(C) } [81,242]}</math>.
  
 +
==Solution 3==
 +
Since the choices are rather wide ranges, we can use approximation to make it easier. Notice that
 +
<cmath>x_{n+1} - x_n = \frac{4-x_n}{x_n+6}</cmath>
 +
And <math>x_0 =5</math>, we know that <math>x_n</math> is a declining sequence, and as it get close to 4 its decline will slow, never falling below 4. So we'll use 4 to approximate <math>x_n</math> in the denominator so that we have a solvable difference equation:
 +
<cmath> x_{n+1} - x_n = \frac{4-x_n}{10}</cmath>
 +
<cmath>x_{n+1} = \frac{9}{10}x_n + \frac{2}{5}</cmath>
 +
Solve it with <math>x_0 = 5</math>, we have
 +
<cmath>x_n = 4 + (\frac{9}{10})^n</cmath>
 +
Now we wish to find <math>n</math> so that
 +
<cmath> (\frac{9}{10})^n \approx \frac{1}{2^{20}}</cmath>
 +
<cmath> n \approx \frac{\log{2^{20}}}{\log{10}-\log9} \approx \frac{20*0.3}{0.05} = 120</cmath>
 +
Since 120 is safely within the range of [81,242], we have the answer.
 +
<math>\boxed{\textbf{(C) } [81,242]}</math>.
  
 +
-Mathdummy
  
The first integer <math>m</math> such that <math>a_{m}\leq\frac{1}{2^{20}}</math> must therefore lie in the interval
+
==Video Solution==
<math>\left[\left\lfloor 20log_{\frac{10}{9}}{2}\right\rfloor+1, \left\lceil 20log_{\frac{11}{10}}{2}\right\rceil\right]</math>
 
  
Both of these can be quickly estimated at c. <math>140</math>, so the answer must be <math>\boxed{C}</math>.
+
Video Solution: https://www.youtube.com/watch?v=Ok7bYOdiF6M
(actual values are <math>132</math> and <math>147</math>)
 
  
 
==See Also==
 
==See Also==

Latest revision as of 21:55, 10 June 2021

The following problem is from both the 2019 AMC 10B #24 and 2019 AMC 12B #22, so both problems redirect to this page.

Problem

Define a sequence recursively by $x_0=5$ and \[x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}\] for all nonnegative integers $n.$ Let $m$ be the least positive integer such that

\[x_m\leq 4+\frac{1}{2^{20}}.\]

In which of the following intervals does $m$ lie?

$\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty)$

Solution 1

We first prove that $x_n > 4$ for all $n \ge 0$, by induction. Observe that \[x_{n+1} - 4 = \frac{x_n^2 + 5x_n + 4 - 4(x_n+6)}{x_n+6} = \frac{(x_n - 4)(x_n+5)}{x_n+6}\] so (since $x_n$ is clearly positive for all $n$, from the initial definition), $x_{n+1} > 4$ if and only if $x_{n} > 4$.

We similarly prove that $x_n$ is decreasing, since \[x_{n+1} - x_n = \frac{x_n^2 + 5x_n + 4 - x_n(x_n+6)}{x_n+6} = \frac{4-x_n}{x_n+6} < 0\]

Now we need to estimate the value of $x_{n+1}-4$, which we can do using the rearranged equation \[x_{n+1} - 4 = (x_n-4)\cdot\frac{x_n + 5}{x_n+6}\] Since $x_n$ is decreasing, $\frac{x_n + 5}{x_n+6}$ is clearly also decreasing, so we have \[\frac{9}{10} < \frac{x_n + 5}{x_n+6} \le \frac{10}{11}\] and \[\frac{9}{10}(x_n-4) < x_{n+1} - 4 \le \frac{10}{11}(x_n-4)\]

This becomes \[\left(\frac{9}{10}\right)^n = \left(\frac{9}{10}\right)^n \left(x_0-4\right) < x_{n} - 4 \le \left(\frac{10}{11}\right)^n \left(x_0-4\right) = \left(\frac{10}{11}\right)^n\] The problem thus reduces to finding the least value of $n$ such that \[\left(\frac{9}{10}\right)^n < x_{n} - 4 \le \frac{1}{2^{20}} \text{  and  }  \left(\frac{10}{11}\right)^{n-1} > x_{n-1} - 4 > \frac{1}{2^{20}}\]

Taking logarithms, we get $n \ln \frac{9}{10} < -20 \ln 2$ and $(n-1)\ln \frac{10}{11} > -20 \ln 2$, i.e.

\[n > \frac{20\ln 2}{\ln\frac{10}{9}} \text{  and  }  n-1 < \frac{20\ln 2}{\ln\frac{11}{10}}\]

As approximations, we can use $\ln\frac{10}{9} \approx \frac{1}{9}$, $\ln\frac{11}{10} \approx \frac{1}{10}$, and $\ln 2\approx 0.7$. These allow us to estimate that \[126 < n < 141\] which gives the answer as $\boxed{\textbf{(C) } [81,242]}$.

Solution 2

The condition where $x_m\leq 4+\frac{1}{2^{20}}$ gives the motivation to make a substitution to change the equilibrium from $4$ to $0$. We can substitute $x_n = y_n + 4$ to achieve that. Now, we need to find the smallest value of $m$ such that $y_m\leq \frac{1}{2^{20}}$ given that $y_0 = 1$.


Factoring the recursion $x_{n+1} = \frac{x_n^2 + 5x_n+4}{x_n + 6}$, we get:

$x_{n+1}=\dfrac{(x_n + 4)(x_n + 1)}{x_n + 6} \Rightarrow y_{n+1}+4=\dfrac{(y_n+8)(y_n+5)}{y_n+10}$

$y_{n+1}+4=\dfrac{y_n^2+13y_n+40}{y_n+10} = \dfrac{y_n^2+9y_n +(4y_n+40)}{y_n+10}$

$y_{n+1}+4=\dfrac{y_n^2+9y_n}{y_n+10} + 4$

$y_{n+1}=\dfrac{y_n^2+9y_n}{y_n+10}$.


Using wishful thinking, we can simplify the recursion as follows:

$y_{n+1} = \frac{y_n^2 + 9y_n + y_n - y_n}{y_n + 10}$

$y_{n+1} = \frac{y_n(y_n + 10) - y_n}{y_n + 10}$

$y_{n+1} = y_n - \frac{y_n}{y_n + 10}$

$y_{n+1} = y_n\left(1 - \frac{1}{y_n + 10}\right)$.


The recursion looks like a geometric sequence with the ratio changing slightly after each term. Notice from the recursion that the $y_n$ sequence is strictly decreasing, so all the terms after $y_0$ will be less than 1. Also, notice that all the terms in sequence will be positive. Both of these can be proven by induction.


With both of those observations in mind, $\frac{9}{10} < 1 - \frac{1}{y_n + 10} \leq \frac{10}{11}$. Combining this with the fact that the recursion resembles a geometric sequence, we conclude that $\left(\frac{9}{10}\right)^n < y_n \leq \left(\frac{10}{11}\right)^n.$

$\frac{9}{10}$ is approximately equal to $\frac{10}{11}$ and the ranges that the answer choices give us are generous, so we should use either $\frac{9}{10}$ or $\frac{10}{11}$ to find a rough estimate for $m$.


Since $\dfrac{1}{2}=0.5$, that means $\frac{1}{\sqrt{2}}=2^{-\frac{1}{2}} \approx 0.7$. Additionally, $\left(\frac{9}{10}\right)^3=0.729$


Therefore, we can estimate that $2^{-\frac{1}{2}} < y_3$.

Raising both sides to the 40th power, we get $2^{-20} < (y_3)^{40}$

But $y_3 = (y_0)^3$, so $2^{-20} < (y_0)^{120}$ and therefore, $2^{-20} < y_{120}$.

This tells us that $m$ is somewhere around 120, so our answer is $\boxed{\textbf{(C) } [81,242]}$.

Solution 3

Since the choices are rather wide ranges, we can use approximation to make it easier. Notice that \[x_{n+1} - x_n = \frac{4-x_n}{x_n+6}\] And $x_0 =5$, we know that $x_n$ is a declining sequence, and as it get close to 4 its decline will slow, never falling below 4. So we'll use 4 to approximate $x_n$ in the denominator so that we have a solvable difference equation: \[x_{n+1} - x_n = \frac{4-x_n}{10}\] \[x_{n+1} = \frac{9}{10}x_n + \frac{2}{5}\] Solve it with $x_0 = 5$, we have \[x_n = 4 + (\frac{9}{10})^n\] Now we wish to find $n$ so that \[(\frac{9}{10})^n \approx \frac{1}{2^{20}}\] \[n \approx \frac{\log{2^{20}}}{\log{10}-\log9} \approx \frac{20*0.3}{0.05} = 120\] Since 120 is safely within the range of [81,242], we have the answer. $\boxed{\textbf{(C) } [81,242]}$.

-Mathdummy

Video Solution

Video Solution: https://www.youtube.com/watch?v=Ok7bYOdiF6M

See Also

2019 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 23
Followed by
Problem 25
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 10 Problems and Solutions
2019 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 21
Followed by
Problem 23
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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