Question

# Statement I For every natural number n≥2 1√1+1√2+⋯+1√n>√n Statement II  For every natural number n≥2                    √n(n+1)<n+1

A
Statement I is true, Statement II is true; and Statement II is correct explanation for Statement I.
B
Statement I is true, Statement II is true; and Statement II is not correct explanation for Statement I.
C
Statement I is true, Statement II is false.
D
Statement I is false, Statement II is true.

Solution

## The correct option is C Statement I is true, Statement II is true; and Statement II is correct explanation for Statement I.For every natural number n, we have  n(n+1)=n2+n<n2+n+n+1 ⇒n(n+1)<(n+1)2 ⇒√n(n+1)<(n+1)  ∀n≥2 ∴ Statement II is true. Also, from above, we have √n<√n+1 ⇒1√n>1√n+1  ∀ n≥2 ⇒1√1>1√2>1√3>⋯>1√n−1>1√n  ∀ n≥2 ⇒1√1>1√n 1√2>1√n 1√3>1√n …  …  … …  …  … 1√n=1√n,∀n≥2 Adding all, we get 1√1+1√2+1√3+⋯+1√n>n√n=√n ∴ Statement I is true.Mathematics

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