The correct option is A 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.