Statement-1: For every natural number `nge2`,
`(1)/(sqrt(1))+(1)/(sqrt(2))+(1)/(sqrt(3))+...+(1)/(sqrt(n))gtsqrt(n)`
Statement-2: For every natural number `nge2,`
`sqrt(n(n+1))ltn+1`

Options

A. Statement-1 is true , Statement-2 is true, Statement-2 is correct explanation for Statement-1
B. Statement-1 is true , Statement-2 is true , Statement-2 is not a correct explanation for Statement-1
C. Statement-1 is true , Statement-2 is false
D. Statement-1 is false , Statement -2 is true .

Explanation

Let `P(n)=(1)/(sqrt(1))+(1)/(sqrt(2))+.....+(1)/(sqrt(n))`
`therefore P(2)=(1)/(sqrt(2))+(1)w/(sqrt(2))=1.707gt sqrt(2)`
Let us assume that
`P(k)=(1)/(sqrt(1))+(1)/(sqrt(2))+.....+(1)/(sqrt(k))gt sqrt(k)` is true for `n=k+1`.
`=(1)/(sqrt(1))+(1)/(sqrt(2))+.....+(1)/(sqrt(k))+(1)/(sqrt(k+1))gt sqrt(k)+(1)/(sqrt(k+1))=sqrt(k(k+1)+1)/(sqrt((k+1)))gt(k+1)/(sqrt((k+1)))" "[therefore sqrt(k(k+1)+1)gtk,forallkge0]`
`therefore P(k+1)gt sqrt((k+1))`
By mathematical induction statement -1 is true , `forall n ge 2` .
Now , let `alpha(n)=sqrt(2(2+1))=sqrt(6)lt 3`
Let us assume that `alpha(k)=sqrt(k(k+1))lt(k+1)` is true
for `n=k+1`
LHS `=sqrt((k+1)(k+2))lt (k+2)" "[therefore (k+1)lt(k+2)]`
By mathematical induction Statement - 2 is true but Statement -2 is not a correct explanation for Statement -1.