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Statement 1: For every natural number `ngeq2` , `1/(sqrt(1))+1/(sqrt(2))+...+1/(sqrt(n))>sqrt(n)` . Statement 2: For every natural number `ngeq2,""n(n+1)<n+1` . (1) Statement 1 is false, Statement `( 2) (3)-2( 4)` is true (6) Statement 1 is true, Statement `( 7) (8)-2( 9)` (10) is true, Statement `( 11) (12)-2( 13)` is a correct explanation for Statement 1 (15) Statement 1 is true, Statement `( 16) (17)-2( 18)` (19) is true; Statement `( 20) (21)-2( 22)` is not a correct explanation for Statement 1. (24) Statement 1 is true, Statement `( 25) (26)-2( 27)` is false.

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2) `sqrt(n(n+1)) = sqrt(n^2 + n)`
`= sqrt((n+1)^2 - (1+n)) < sqrt((n+1)^2)`
`sqrt(n^2 + n) < n+1`
1)`P(n) = 1/sqrt1 + 1/sqrt2 + ...... + 1/sqrt n > sqrt n`
`P(n+1) = 1/sqrt1 + 1/sqrt2 + .... 1/sqrtn + 1/sqrt(n+1) > sqrt(n+1)`
`P(n+1) > sqrtn + 1/ sqrt(n+1)`
`P(n+1) > ( sqrt(n(n+1)) + 1)/(sqrt(n+1))`
`sqrt(n(n+1)) > n`
`sqrt(n(n+1)) + 1 > n+1`
`(sqrt(n(n+1)) + 1)/(sqrt(n+1)) > (n+1)/sqrt(n+1) `
`(sqrt(n(n+1)) + 1)/( sqrt(n+1)) > sqrt(n+1)`
`P(n+1) > sqrt(n+1)`
option 3 is correct

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