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
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#.