Can You Simplify?

Algebra Level pending

How many positive integers n n such that n n is less than 1000, make the following expression a natural number as well?

n + n + n + ⋯ \sqrt{ n + \sqrt{n + \sqrt{n + \cdots}}}

This is based off the Infinitely Nested Radical Problem


The answer is 31.

Clive Chen by Clive Chen , 25, USA

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1 solution

Let n + n + n + ⋯ = k \sqrt{ n+\sqrt{n+\sqrt{n +\cdots}}}=k with k k is a positive integer.

We have: n + k = k \sqrt{n+k}=k , which implies n = k 2 − k n=k^2-k .

For posive integer k k , we have 0 < k 2 − k < 1000 0<k^2-k<1000 when 2 ≤ k ≤ 32 2\le k\le 32 .

So the final answer is 31 \boxed{31} .

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