# 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?

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**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.