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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