What is the smallest positive integer n such that √ n-√n-1<0.01?

1978 AMC 12 Problem 18

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18.

What is the smallest positive integer

n

such that

\sqrt n-\sqrt{n-1}\lt0.01?

2499

2500

2501

10{,}000

There is no such integer.

Rationalize

\sqrt n-\sqrt{n-1}

Compare the resulting denominator with

100

near

n=2500

Rationalizing gives

\sqrt n-\sqrt{n-1} =\frac1{\sqrt n+\sqrt{n-1}}.

For

n=2500,

the denominator is

50+\sqrt{2499}\lt100,

so the difference exceeds

0.01.

For

n=2501,

the denominator is

\sqrt{2501}+50\gt100,

so the difference is less than

0.01.

The denominator increases with

n,

making

2501

the least such integer.

Therefore, the correct answer is C.