Difference between revisions of "2021 Fall AMC 12B Problems/Problem 18"
(Created page with "Set <math>u_0 = \frac{1}{4}</math>, and for <math>k \ge 0</math> let <math>u_{k+1}</math> be determined by the recurrence <cmath>u_{k+1} = 2u_k - 2u_k^2.</cmath> This sequenc...") |
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| + | ==Problem== | ||
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Set <math>u_0 = \frac{1}{4}</math>, and for <math>k \ge 0</math> let <math>u_{k+1}</math> be determined by the recurrence <cmath>u_{k+1} = 2u_k - 2u_k^2.</cmath> | Set <math>u_0 = \frac{1}{4}</math>, and for <math>k \ge 0</math> let <math>u_{k+1}</math> be determined by the recurrence <cmath>u_{k+1} = 2u_k - 2u_k^2.</cmath> | ||
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<math>(\textbf{A})\: 10\qquad(\textbf{B}) \: 87\qquad(\textbf{C}) \: 123\qquad(\textbf{D}) \: 329\qquad(\textbf{E}) \: 401</math> | <math>(\textbf{A})\: 10\qquad(\textbf{B}) \: 87\qquad(\textbf{C}) \: 123\qquad(\textbf{D}) \: 329\qquad(\textbf{E}) \: 401</math> | ||
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| + | ==Solution== | ||
Revision as of 17:48, 24 November 2021
Problem
Set 
, and for 
let 
be determined by the recurrence ![\[u_{k+1} = 2u_k - 2u_k^2.\]](http://latex.artofproblemsolving.com/4/d/2/4d293888df7007ec4ada4b226a235ad82caf9d9f.png)
This sequence tends to a limit; call it 
. What is the least value of 
such that ![\[|u_k-L| \le \frac{1}{2^{1000}}?\]](http://latex.artofproblemsolving.com/6/c/d/6cdccfccc047b6c8985ceea4d5d27e3bc9e4bab3.png)
