The Least Radical Difference

Algebra Level 4

Find the least positive integral value of x x such that x − x − 1 < 1 2016 . \sqrt x - \sqrt{x-1} < \dfrac1{2016} \; .


The answer is 1016065.

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Choi Chakfung by choi chakfung , 28, Hong Kong

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1 solution

Otto Bretscher
Apr 3, 2016

We have 1 2 x − 1 > x − x − 1 > 1 2 x \frac{1}{2\sqrt{x-1}}>\sqrt{x}-\sqrt{x-1}>\frac{1}{2\sqrt{x}} . We want 1 2 x − 1 = 1 2016 \frac{1}{2\sqrt{x-1}}=\frac{1}{2016} so x = 100 8 2 + 1 = 1016065 x=1008^2+1=\boxed{1016065}

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Moderator note:

The inequality in the first line follows from considering x − x − 1 = 1 x + x − 1 \sqrt{x} - \sqrt{x-1} = \frac{ 1}{ \sqrt{x} + \sqrt{ x - 1 } } .

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