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Math Help - Induction Proof

  1. #1
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    Induction Proof

    Prove that 2^{\frac{1}{n}} <= 1 + \frac{1}{n}.
    for n in the set of all positive numbers.

    Base case is obvious.
    So, tried using n = k, and adding k+1.
    But then I get a load of terms and it gets too messy to handle.

    Any advice?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by BlackBlaze View Post
    Prove that 2^{\frac{1}{n}} <= 1 + \frac{1}{n}.
    for n in the set of all positive numbers.

    Base case is obvious.
    So, tried using n = k, and adding k+1.
    But then I get a load of terms and it gets too messy to handle.

    Any advice?
    Hint
    2^{\frac{1}{n}}\leqslant 1+\frac{1}{n}\Leftrightarrow 2\leqslant\left(1+\frac{1}{n}\right)^n. That's easier to work with.
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