← JP Morgan Interview Insights
The trick is recognizing that the nested part is just x again, so you get x = sqrt(2 + x), square both sides, and solve the quadratic.
Recognize that the infinite radical satisfies the equation x = sqrt(2 + x). Square both sides to get x^2 = 2 + x, then solve the quadratic x^2 - x - 2 = 0 to find x = 2 or x = -1. Since x is a square root, it must be non-negative, so x = 2.
Pro tip: After finding the solution, briefly verify convergence by showing the sequence of partial radicals is increasing and bounded above by 2, which demonstrates rigor and prevents overlooking extraneous roots.
Observe that the expression inside the outer square root is 2 plus the same infinite radical, so the entire expression equals sqrt(2 + x).
Write x = sqrt(2 + x) to capture the infinite nesting.
Square both sides to obtain x^2 = 2 + x, rearrange to x^2 - x - 2 = 0, and factor to (x - 2)(x + 1) = 0, giving x = 2 or x = -1.
Since x is defined as a square root, it must be non-negative, so discard x = -1 and conclude x = 2.
Show that the sequence of partial radicals converges to 2 by proving it is increasing and bounded above by 2, ensuring the solution is valid.
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