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Math Help - quick question about inductive proof

  1. #1
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    quick question about inductive proof

    Prove that:  3^n < n! n is an integer greater than 6.

    Basic step:
    P(7): 3^7 < 7!
    2187 < 5040

    Inductive Steps:
    hypothesis: P(k): 3^k < k!
    conclusion: P(k+1): 3^{k+1} < (k+1)!

    Proof: 3^(k+1) = 3.3^k < 3.k! < (k+1)k! = (k+1)!


    why is this one: (k+1)k! equal to (k+1)!?

    Is there any better way to prove it?
    Last edited by zpwnchen; October 17th 2009 at 09:28 AM.
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  2. #2
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    If you have more than one character in an exponent, set off the whole exponent in braces.
    [tex]3^{(k+1)}[/tex] gives 3^{(k+1)} instead of 3^(k+1).
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  3. #3
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    Quote Originally Posted by zpwnchen View Post
    Prove that:  3^n < n! n is an integer greater than 6.

    Basic step:
    P(7): 3^7 < 7!
    2187 < 5040

    Inductive Steps:
    hypothesis: P(k): 3^k < k!
    conclusion: P(k+1): 3^(k+1) < (k+1)!

    Proof: 3^(k+1) = 3.3^k < 3.k! < (k+1)k! = (k+1)!


    why is this one: (k+1)k! equal to (k+1)!?

    Is there any better way to prove it?
    k! = 1\cdot 2 \cdot 3 \cdot ... \cdot k
    (k+1)! = 1\cdot 2 \cdot 3 \cdot ... \cdot k \cdot (k+1)
    (k+1)k! = (k+1)(1\cdot 2 \cdot 3 \cdot ... \cdot k) = 1\cdot 2 \cdot 3 \cdot ... \cdot k \cdot (k+1) = (k+1)!
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