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Source: 12B #18

November 17, 2021
AMCAMC 12AMC 12 B

Problem Statement

Set u0=14,u_0 = \frac{1}{4}, and for k≥0k \geq 0 let uk+1u_{k+1} be determined by the recurrence uk+1=2uk−2uk2.u_{k+1} = 2u_k - 2u_k^2. This sequence tends to a limit, call it L.L. What is the least value of kk such that ∣uk−L∣≤121000?|u_k - L| \leq \frac{1}{2^{1000}}?
<spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 97<spanclass=′latex−bold′>(C)</span> 253<spanclass=′latex−bold′>(D)</span> 329<spanclass=′latex−bold′>(E)</span> 401<span class='latex-bold'>(A)</span>\ 10 \qquad<span class='latex-bold'>(B)</span>\ 97 \qquad<span class='latex-bold'>(C)</span>\ 253 \qquad<span class='latex-bold'>(D)</span>\ 329 \qquad<span class='latex-bold'>(E)</span>\ 401
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