An Infinitely Nested Radical

Number Theory Level 3

It can be shown that for any positive integer \(n\), the infinitely nested radical expression \[ \sqrt{ n + \sqrt{n + \sqrt{n + \cdots}}} \] equals a finite number. What is the largest positive integer \( n \le 999 \) 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 \( \sqrt{ 3 + \sqrt{5}}.\)
  • An infinitely nested radical expression is one in which the radicals continue to an infinite extent.

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