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Math Help - proof by induction

  1. #1
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    proof by induction

    Prove by induction that 3^n>n^3 where n>3 please.
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  2. #2
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by Rubberduckzilla View Post
    Prove by induction that 3^n>n^3 where n>3 please.

    Verify n = 4 : 3^4 > 4^3 sure enough, true.

    Assume true for n = k - 1 : 3^{k-1} > (k-1)^3

    Prove for n = k: 3^k > k^3


    Left Side =  3^k = 3 3^{k-1} > 3 (k-1)^3  > k^3

    the last inequality is true because:

    \frac{1}{3} < (\frac{k-1}{k})^3

    \frac{1}{3} < (1 - \frac{1}{k})^3

    When k = 4, we have 1/3 < 9/16
    When k = 5, we have 1/3 < 16/25

    as k goes to infinity: we have (1/3) < 1

    Good luck!!
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