An Infinitely Nested Radical

It can be shown that for any positive integer nn, the infinitely nested radical expression n+n+n+ \sqrt{ n + \sqrt{n + \sqrt{n + \cdots}}} equals a finite number. What is the largest positive integer n999 n \le 999 such that this expression is equal to a positive integer?

Details and assumptions:

  • A nested radical expression is one which contains a radical inside another, as in 3+5. \sqrt{ 3 + \sqrt{5}}.
  • An infinitely nested radical expression is one in which the radicals continue to an infinite extent.
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