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Real Analysis on AMC

Source: 2019 AMC 12B #22

February 14, 2019
real analysis2019 AMC 12BAMCAMC 12AMC 12 B2019 AMC 10B

Problem Statement

Define a sequence recursively by x0=5x_0=5 and xn+1=xn2+5xn+4xn+6x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6} for all nonnegative integers n.n. Let mm be the least positive integer such that xm≤4+1220.x_m\leq 4+\frac{1}{2^{20}}. In which of the following intervals does mm lie?
<spanclass=′latex−bold′>(A)</span>[9,26]<spanclass=′latex−bold′>(B)</span>[27,80]<spanclass=′latex−bold′>(C)</span>[81,242]<spanclass=′latex−bold′>(D)</span>[243,728]<spanclass=′latex−bold′>(E)</span>[729,∞]<span class='latex-bold'>(A) </span> [9,26] \qquad<span class='latex-bold'>(B) </span> [27,80] \qquad<span class='latex-bold'>(C) </span> [81,242]\qquad<span class='latex-bold'>(D) </span> [243,728] \qquad<span class='latex-bold'>(E) </span> [729,\infty]
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