Problem: What is the smallest positive integer nnn such that n−n−1<.01?\sqrt{n}-\sqrt{n-1}<.01?n−n−1<.01?
Answer Choices:
A. 249924992499
B. 250025002500
C. 250125012501
D. 100001000010000
E. There is no such integer Solution:
n−n−1<1100\sqrt{n}-\sqrt{n-1}<\dfrac{1}{100}n−n−1<1001;
n+n−1=1n−n−1>1002500+2499<1002501+2500>100n=2501.\begin{aligned} & \sqrt{n}+\sqrt{n-1}=\dfrac{1}{\sqrt{n}-\sqrt{n-1}}>100 \\ & \sqrt{2500}+\sqrt{2499}<100 \\ & \sqrt{2501}+\sqrt{2500}>100 \\ & \quad n=2501 . \end{aligned} n+n−1=n−n−11>1002500+2499<1002501+2500>100n=2501.