Published by:
CGP EDU Academic Team
Published on: August 13, 2026
Show that the square roots of two successive natural numbers greater than N 2 differ by less than 
.
Text Solution
Verified by ExpertsThe correct answer is:
CHECK THE SOLUTION.
Sol. Let n be a natural number greater than N 2 . Consider the function f(x) = 
defined on the interval [n, n + 1]. Clearly, f(x) satisfies all the conditions of Lagrange's mean value theorem on [n, n + 1]. So, there exists c ∈ (n, n + 1) such that
f ′ = 
⇒ 
= 
– 
Now, n > N 2 and c ∈ (n, n + 1)
∴ c > N 2 ⇒ 
> N
⇒ 2 
> 2N
⇒ 
< 
⇒ 
– 
< 
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