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Difference between revisions of "2021 Fall AMC 12B Problems/Problem 18"

m (→‎Solution 2)
(Swapped solutions, as Sol 1 is a popular approach. Hope all users are ok with this ...)
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==Solution 1==
 
==Solution 1==
 +
 +
Note that terms of the sequence <math>(u_k)</math> lie in the interval <math>\left(0,\frac12\right),</math> strictly increasing.
 +
 +
Since the sequence <math>(u_k)</math> tends to the limit <math>L,</math> we set <math>u_{k+1}=u_k=L>0.</math>
 +
 +
The given equation becomes <cmath>L=2L-2L^2,</cmath> from which <math>L=\frac12.</math>
 +
 +
The given inequality becomes <cmath>\frac12-\frac{1}{2^{1000}} \leq u_k \leq \frac12+\frac{1}{2^{1000}},</cmath> and we only need to consider <math>\frac12-\frac{1}{2^{1000}} \leq u_k.</math>
 +
 +
We have
 +
<cmath>\begin{alignat*}{8}
 +
u_0 &= \phantom{1}\frac14 &&= \frac{2^1-1}{2^2}, \\
 +
u_1 &= \phantom{1}\frac38 &&= \frac{2^2-1}{2^3}, \\
 +
u_2 &= \ \frac{15}{32} &&= \frac{2^4-1}{2^5}, \\
 +
u_3 &= \frac{255}{512} &&= \frac{2^8-1}{2^9}, \\
 +
& \phantom{1111} \vdots
 +
\end{alignat*}</cmath>
 +
By induction, it can be proven that <cmath>u_k=\frac{2^{2^k}-1}{2^{2^k+1}}=\frac12-\frac{1}{2^{2^k+1}}.</cmath>
 +
We substitute this into the inequality, then solve for <math>k:</math>
 +
<cmath>\begin{align*}
 +
\frac12-\frac{1}{2^{1000}} &\leq \frac12-\frac{1}{2^{2^k+1}} \\
 +
-\frac{1}{2^{1000}} &\leq -\frac{1}{2^{2^k+1}} \\
 +
2^{1000} &\leq 2^{2^k+1} \\
 +
1000 &\leq 2^k+1.
 +
\end{align*}</cmath>
 +
Therefore, the least such value of <math>k</math> is <math>\boxed{\textbf{(A)}\: 10}.</math>
 +
 +
~MRENTHUSIASM
 +
 +
==Solution 2==
  
 
If we list out the first few values of <math>k</math>, we get the series <math>\frac{1}{4}, \frac{3}{8}, \frac{15}{32}, \frac{255}{512}</math>, which seem to always be a negative power of <math>2</math> away from <math>\frac{1}{2}</math>. We can test this out by setting <math>u_k</math> to <math>\frac{1}{2}-\frac{1}{2^{n_k}}</math>.  
 
If we list out the first few values of <math>k</math>, we get the series <math>\frac{1}{4}, \frac{3}{8}, \frac{15}{32}, \frac{255}{512}</math>, which seem to always be a negative power of <math>2</math> away from <math>\frac{1}{2}</math>. We can test this out by setting <math>u_k</math> to <math>\frac{1}{2}-\frac{1}{2^{n_k}}</math>.  
Line 37: Line 67:
  
 
~ConcaveTriangle
 
~ConcaveTriangle
−
 
−
==Solution 2==
 
−
 
−
Note that terms of the sequence <math>(u_k)</math> lie in the interval <math>\left(0,\frac12\right),</math> strictly increasing.
 
−
 
−
Since the sequence <math>(u_k)</math> tends to the limit <math>L,</math> we set <math>u_{k+1}=u_k=L>0.</math>
 
−
 
−
The given equation becomes <cmath>L=2L-2L^2,</cmath> from which <math>L=\frac12.</math>
 
−
 
−
The given inequality becomes <cmath>\frac12-\frac{1}{2^{1000}} \leq u_k \leq \frac12+\frac{1}{2^{1000}},</cmath> and we only need to consider <math>\frac12-\frac{1}{2^{1000}} \leq u_k.</math>
 
−
 
−
We have
 
−
<cmath>\begin{alignat*}{8}
 
−
u_0 &= \phantom{1}\frac14 &&= \frac{2^1-1}{2^2}, \\
 
−
u_1 &= \phantom{1}\frac38 &&= \frac{2^2-1}{2^3}, \\
 
−
u_2 &= \ \frac{15}{32} &&= \frac{2^4-1}{2^5}, \\
 
−
u_3 &= \frac{255}{512} &&= \frac{2^8-1}{2^9}, \\
 
−
& \phantom{1111} \vdots
 
−
\end{alignat*}</cmath>
 
−
By induction, it can be proven that <cmath>u_k=\frac{2^{2^k}-1}{2^{2^k+1}}=\frac12-\frac{1}{2^{2^k+1}}.</cmath>
 
−
We substitute this into the inequality, then solve for <math>k:</math>
 
−
<cmath>\begin{align*}
 
−
\frac12-\frac{1}{2^{1000}} &\leq \frac12-\frac{1}{2^{2^k+1}} \\
 
−
-\frac{1}{2^{1000}} &\leq -\frac{1}{2^{2^k+1}} \\
 
−
2^{1000} &\leq 2^{2^k+1} \\
 
−
1000 &\leq 2^k+1.
 
−
\end{align*}</cmath>
 
−
Therefore, the least such value of <math>k</math> is <math>\boxed{\textbf{(A)}\: 10}.</math>
 
−
 
−
~MRENTHUSIASM
 
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2021 Fall|ab=B|num-a=19|num-b=17}}
 
{{AMC12 box|year=2021 Fall|ab=B|num-a=19|num-b=17}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 02:31, 19 March 2022

Problem

Set $u_0 = \frac{1}{4}$, and for $k \ge 0$ let $u_{k+1}$ be determined by the recurrence \[u_{k+1} = 2u_k - 2u_k^2.\]

This sequence tends to a limit; call it $L$. What is the least value of $k$ such that \[|u_k-L| \le \frac{1}{2^{1000}}?\]

$\textbf{(A)}\: 10\qquad\textbf{(B)}\: 87\qquad\textbf{(C)}\: 123\qquad\textbf{(D)}\: 329\qquad\textbf{(E)}\: 401$

Solution 1

Note that terms of the sequence $(u_k)$ lie in the interval $\left(0,\frac12\right),$ strictly increasing.

Since the sequence $(u_k)$ tends to the limit $L,$ we set $u_{k+1}=u_k=L>0.$

The given equation becomes \[L=2L-2L^2,\] from which $L=\frac12.$

The given inequality becomes \[\frac12-\frac{1}{2^{1000}} \leq u_k \leq \frac12+\frac{1}{2^{1000}},\] and we only need to consider $\frac12-\frac{1}{2^{1000}} \leq u_k.$

We have \begin{alignat*}{8} u_0 &= \phantom{1}\frac14 &&= \frac{2^1-1}{2^2}, \\ u_1 &= \phantom{1}\frac38 &&= \frac{2^2-1}{2^3}, \\ u_2 &= \ \frac{15}{32} &&= \frac{2^4-1}{2^5}, \\ u_3 &= \frac{255}{512} &&= \frac{2^8-1}{2^9}, \\ & \phantom{1111} \vdots \end{alignat*} By induction, it can be proven that \[u_k=\frac{2^{2^k}-1}{2^{2^k+1}}=\frac12-\frac{1}{2^{2^k+1}}.\] We substitute this into the inequality, then solve for $k:$ \begin{align*} \frac12-\frac{1}{2^{1000}} &\leq \frac12-\frac{1}{2^{2^k+1}} \\ -\frac{1}{2^{1000}} &\leq -\frac{1}{2^{2^k+1}} \\ 2^{1000} &\leq 2^{2^k+1} \\ 1000 &\leq 2^k+1. \end{align*} Therefore, the least such value of $k$ is $\boxed{\textbf{(A)}\: 10}.$

~MRENTHUSIASM

Solution 2

If we list out the first few values of $k$, we get the series $\frac{1}{4}, \frac{3}{8}, \frac{15}{32}, \frac{255}{512}$, which seem to always be a negative power of $2$ away from $\frac{1}{2}$. We can test this out by setting $u_k$ to $\frac{1}{2}-\frac{1}{2^{n_k}}$.

Now, we get \begin{align*} u_{k+1} &= 2\cdot\left(\frac{1}{2}-\frac{1}{2^{n_{k}}}\right)-2\cdot\left(\frac{1}{2}-\frac{1}{2^{n_{k}}}\right)^2 \\ &= 1-\frac{1}{2^{n_k - 1}}-2\cdot\left(\frac{1}{4}-\frac{1}{2^{n_k}}+\frac{1}{2^{2 \cdot n_k}}\right)\\ &= 1-\frac{1}{2^{n_k - 1}}-\frac{1}{2}+\frac{1}{2^{n_k-1}}-\frac{1}{2^{2 \cdot n_k-1}} \\ &= \frac{1}{2}-\frac{1}{2^{2 \cdot n_k-1}}. \end{align*} This means that this series approaches $\frac{1}{2}$, as the second term is decreasing. In addition, we find that $n_{k+1}=2 \cdot n_k-1$.

We see that $n_k$ seems to always be $1$ above a power of $2$. We can prove this using induction.

Claim

$n_k = 2^k+1$

Base case

We have $n_0=2^0+1$, which is true.

Induction Step

Assuming that the claim is true, we have $n_{k+1}=2 \cdot 2^k+2-1=2^{k+1}+1$.

It follows that $n_{10}=2^{10}+1>1000$, and $n_9=2^9+1<1000$. Therefore, the least value of $k$ would be $\boxed{\textbf{(A)}\: 10}$.

~ConcaveTriangle

See Also

2021 Fall AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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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