Question

What is the smallest positive integer n?

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

Answer 1

`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`.