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Statement 1: For every natural number n...

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)

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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`
...
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Knowledge Check

  • 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

    A
    1
    B
    2
    C
    3
    D
    4

  • For every natural number n (n + 1) is always

    A
    Even
    B
    Odd
    C
    Multiple of 3
    D
    Multiple of 4

  • For every natural number n,n (n^(2)-1) is divisible by

    A
    4
    B
    6
    C
    10
    D
    None of these