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.
by
Calvin Lin