What is the smallest positive integer n?

What is the smallest positive integer n such that \sqrt{n} - \sqrt{n-1} < 0.01?

1 Answer
NickTheTurtle
May 13, 2017

n=2501

Explanation:

Find when sqrt(x)-sqrt(x-1)=0.01. Then, n would be the smallest positive integer greater than x.

(sqrt(x)-sqrt(x-1))*(sqrt(x)+sqrt(x-1))=0.01(sqrt(x)+sqrt(x-1)), or 1=0.01(sqrt(x)+sqrt(x-1)).

This means that {(sqrt(x)+sqrt(x-1)=100),(sqrt(x)-sqrt(x-1)=0.01):}. Add both equations to get 2sqrt(x)=100.01. Find x to get (100.01/2)^2. This means that 2500< x<2501. Thus, the smallest positive integer n greater than x is 2501.

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