Difference between revisions of "2019 AMC 10B Problems/Problem 24"
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| − | + | Define a sequence recursively by <math>x_0=5</math> and | |
| + | <cmath>x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}</cmath>for all nonnegative integers <math>n.</math> Let <math>m</math> be the least positive integer such that | ||
| + | <cmath>x_m\leq 4+\frac{1}{2^{20}}.</cmath>In which of the following intervals does <math>m</math> lie? | ||
| + | |||
| + | <math>\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty]</math> | ||
Revision as of 12:52, 14 February 2019
Define a sequence recursively by 
and
![\[x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}\]](http://latex.artofproblemsolving.com/9/3/4/934a32343352988b8d8328a7ef93c7553a1ce642.png)
for all nonnegative integers 
Let 
be the least positive integer such that
![\[x_m\leq 4+\frac{1}{2^{20}}.\]](http://latex.artofproblemsolving.com/e/8/8/e889e3b4b1d11c0fd3ce0eda55858624351b69fd.png)
In which of the following intervals does lie?
![$\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty]$](http://latex.artofproblemsolving.com/6/c/2/6c2f679a426a4923a52cc6f0de03e28c75165307.png)
