2 ≤ n, n ∈ N. prove ((n+1)/2)ⁿ > n!?

2 Answers

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  • 2 years ago
    Best Answer

    Using the AM-GM inequality :

    (1+2+3+...+n)/n > [1*2*3*...n]^(1/n)

    Equality does not hold because 1 ≠ 2 ≠ 3 ≠ etc. for n >= 2.

    LHS = [n(n+1)/2][1/n] = (n+1)/2

    RHS = [n!]^(1/n)

    Therefore

    [(n+1)/2]^n > n!

  • Anonymous
    2 years ago

    (1 + 2 + 3 + ... + n)/n > ⁿ√(n!)

    [(n(n + 1))/2]/n > √(n!)

    (n + 1)/2 > ⁿ√(n!)

    ((n + 1)/2)ⁿ > n!          edit.

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