Question

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

Answer 1

√n - √(n - 1) < 0.01

√n < 0.01 + √(n - 1)

Take squares on both sides and simplify:

n < 0.0001 + 0.02 √(n - 1) + (n - 1)

0.9999 < 0.02 √(n - 1)

49.995 < √(n - 1)

Take squares on both sides again:

2499.5 < n - 1

2500.5 < n

So we must have n ≥ 2501 for the inequality to hold.