# Points on a curve

Algebra Level 5

On the $$xy$$-coordinate plane, $$A_{0}=(0,0), A_{1}, A_{2}, A_{3}\cdots$$... are points that lie on the $$x$$-axis and $$B_{1}, B_{2}, B_{3}, B_{4}\cdots$$ are points that lie on the curve $$y=\sqrt{x}$$ such that for all natural numbers $$k$$, the triangle formed by $$A_{k-1}$$, $$A_{k}$$ and $$B_{k}$$ is equilateral. Joel randomly picks a natural number $$l$$. What is the probability that $$A_{l}$$ has integer coordinates? Give your answer to 3 decimal places.

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