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Difference between revisions of "2021 Fall AMC 12B Problems/Problem 18"
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==Solution 2==
 
==Solution 2==
  
−
Note that all terms of the sequence <math>(u_k)</math> lie in the interval <math>\left(0,\frac12\right),</math> strictly increasing.
+
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>
+
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 &= \frac14 &&= \frac{2^1-1}{2^2}, \\
 +
u_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}, \\
 +
& \ \vdots
 +
\end{alignat*}</cmath>
  
 
==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 21:57, 17 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

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

Solution 2

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 &= \frac14 &&= \frac{2^1-1}{2^2}, \\ u_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}, \\ & \ \vdots \end{alignat*}

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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

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