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Omni-MATH / Consider a sequence x_(n) such that x₁=x₂=1,x₃=(2)/(3). Suppose that x_(n)=(x_(n−1)²x_(n−2))/(2x_(n−2)²−x_(n−1)x_(n−3)) for all n≥4. Find the least n such that x_(n)≤(1)/(10⁶).

Problem

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problem

Consider a sequence xnx_{n} such that x1=x2=1,x3=23x_{1}=x_{2}=1, x_{3}=\frac{2}{3}. Suppose that xn=xn−12xn−22xn−22−xn−1xn−3x_{n}=\frac{x_{n-1}^{2} x_{n-2}}{2 x_{n-2}^{2}-x_{n-1} x_{n-3}} for all n≥4n \geq 4. Find the least nn such that xn≤1106x_{n} \leq \frac{1}{10^{6}}.
Plain-text mathematical notation (without MathML)
Consider a sequence x_(n) such that x₁=x₂=1,x₃=(2)/(3). Suppose that x_(n)=(x_(n−1)²x_(n−2))/(2x_(n−2)²−x_(n−1)x_(n−3)) for all n≥4. Find the least n such that x_(n)≤(1)/(10⁶).
Original LaTeX notation
Consider a sequence $x_{n}$ such that $x_{1}=x_{2}=1, x_{3}=\frac{2}{3}$. Suppose that $x_{n}=\frac{x_{n-1}^{2} x_{n-2}}{2 x_{n-2}^{2}-x_{n-1} x_{n-3}}$ for all $n \geq 4$. Find the least $n$ such that $x_{n} \leq \frac{1}{10^{6}}$.

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Source and history

task_id
0f68fd40-551f-552e-bcb4-fb6e82c86338
task_key
test--0f68fd40-551f-552e-bcb4-fb6e82c86338
task_revision_id
1
config / split
/ test

Official source

initial import