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Statement -1 For every natural number n ...

Statement -1 For every natural number `n ge 2(1)/(sqrt(1))+(1)/(sqrt(2))+.....+(1)/(n)ge sqrt(n)`,
Statement -2 For every natural number `n ge 2 sqrt(n(n+1))lt n+1`

A

1

B

2

C

3

D

4

Text Solution

Verified by Experts

The correct Answer is:
B
Using
`(a_(1)^(m)+a_(2)^(m)+...+a_(n)^(m))/(n)gt((a_(1)+a_(2)+...+a_(n))/(n))^(m)` for `mlt0` or `mgt1,`
we have
`((1)/(sqrt(1))+(1)/(sqrt(2))+(1)/(sqrt(3))+...+(1)/(sqrt(n)))/(n)gt((1+2+...n)/(n))^(-1//2)`
`implies" "(1)/(sqrt(1))+(1)/(sqrt(2))+(1)/(sqrt(3))+...+(1)/(sqrt(n))gtnsqrt((2)/(n+1))`
`implies" "(1)/(sqrt(1))+(1)/(sqrt(2))+(1)/(sqrt(3))+...+(1)/(sqrt(n))gtsqrt(n)sqrt((2n)/(n+1))`
`implies" "(1)/(sqrt(1))+(1)/(sqrt(2))+(1)/(sqrt(3))+...+(1)/(sqrt(n))gtsqrt(n)`
`" "[because 2ngtn+1" for all "nle2]`
So, statement-1 is true.
We have,
`n(n+1)lt(n+1)(n+1)" for all "nge2`
`implies" "sqrt(n(n+1))ltn+1" for all "nge2`
So, statement-2 is true.
Clearly, statement-2 is not a correct explanation for statement-1.
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