I am not going to test all of these!

Number Theory Level 2

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

If n = 420 , n = 420, then the expression above simplifies to an integer.

What is the smallest value of n n which is larger than 420 such that the same expression above simplifies to an integer as well?


The answer is 462.

Munem Shahriar by Munem Shahriar , 18, Bangladesh

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1 solution

Stephen Mellor
Dec 10, 2017

Let x = n + n + n + ⋯ x = \sqrt {n + \sqrt { n + \sqrt { n+ \sqrt{ \cdots} } } }

We can then say that x = n + x x = \sqrt {n + x }

Simplifying we get n = x 2 − x n = x^2 - x

Or n = x ( x − 1 ) n = x(x - 1)

Therefore, as we are looking for integer values of x x , and we know that 420 = 21 × 20 420 = 21 \times 20 , the next solution will be 22 × 21 = 462 22 \times 21 = \boxed{462}

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