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Math Help - Factorial Proof Help

  1. #1
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    Factorial Proof Help

    I am trying to prove the equation for any integer (k) greater or equal to 0, that

    2^(k+1) < (k+3)!

    I am not sure how to expand the factorial to show that the equation will always be true?

    Any Help Would Be Greatly Appreciated
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  2. #2
    MHF Contributor chisigma's Avatar
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    Re: Factorial Proof Help

    Quote Originally Posted by doleary22 View Post
    I am trying to prove the equation for any integer (k) greater or equal to 0, that

    2^(k+1) < (k+3)!

    I am not sure how to expand the factorial to show that the equation will always be true?

    Any Help Would Be Greatly Appreciated
    If we write...

    2^{k+1}= \prod_{i=1}^{k+1} a_{i} (1)

    (k+3)!= \prod_{i=1}^{k+2} b_{i} (2)

    ... You observe that the product (2) has k+2 factors and the product (1) k+1 factors and for i=1,2,...,k+1 is a_{i}\le b_{i}, so that the conclusion is obvious...

    Kind regards

    \chi \sigma
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  3. #3
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    Re: Factorial Proof Help

    I am still a little confused because I am not familiar with the notation. Is there any other way to solve the problem?
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  4. #4
    MHF Contributor chisigma's Avatar
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    Re: Factorial Proof Help

    Quote Originally Posted by doleary22 View Post
    I am still a little confused because I am not familiar with the notation. Is there any other way to solve the problem?
    A more elementary concept probably doesn't exist...

    2^{k+1}= 2 \cdot 2 \cdot ... \cdot 2\ (k+1\ \text{times}) (1)

    (k+3)!= 2 \cdot 3 \cdot ... \cdot (k+2)\ \cdot (k+3) (2)

    Do You understand now?...

    Kind regards

    \chi \sigma
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  5. #5
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    Re: Factorial Proof Help

    I understand how to solve each side of the equation, but what I am confused about is how did you determine the factors in your first response, and then how do I compare those two products to show that the one will always be larger than the other?
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  6. #6
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    Re: Factorial Proof Help

    Hello, doleary22!

    Here is an observation.
    I hope you can modify it into a proof.


    \text{Prove that, for any integer }k \ge 0\!:\;\;2^{k+1} \:<\: (k+3)!

    \text{We have: }\;\underbrace{2\cdot2\cdot2\,\cdots\,2} \quad ^<_>\quad  1\cdot2\cdot\underbrace{3\cdot4\,\cdots\,(k+3)}
    . . . . . . . . . ^{k+1\text{ factors}} \qquad\qquad\qquad \quad ^{k+1\text{ factors } >\: 2}

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  7. #7
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    Re: Factorial Proof Help

    I am still a little confused. I know that the statement is correct, but I just can't seem to wrap my head around a logical way to write down a proof.
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