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Homework Help: Inequality Mathematical Induction

  1. Jul 11, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove the Inequality for the indicated integer values of n.


    2. Relevant equations

    3. The attempt at a solution

    For n=4 the formula is true because


    Assume the n=k


    Now I need to prove the equation for k+1

    I can multiply both sides by 2 and have


    Is this what you would do next? I'm not quite sure what to do past this point.

  2. jcsd
  3. Jul 11, 2010 #2


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    Why would you multiply by 2?
    It's not 2k! you want to make a statement about, but (k + 1)! = (k + 1) k!
  4. Jul 11, 2010 #3
    I dont know. When dealing with regular equations you would just set k=(k+1) but that's not the case here I think.

    Wouldn't you be trying to prove that [tex](k+1)!>2^{k+1}[/tex]? Why would I be making a statement about '(k + 1)! = (k + 1) k!'? Where is the 2 raised to the power of (k+1) and why is there 2 factorials when there isn't in the original problem?
  5. Jul 11, 2010 #4


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    Sorry, you are approaching the problem from a different angle than I expected.

    Once you have 2 (k!) > 2k + 1, you only need to prove that (k + 1)! > 2 k! to be done with it.

    The hint I gave, that (k + 1)! is equal to (k + 1) times k!, is still useful.
  6. Jul 11, 2010 #5
    Ahh, so you could rewrite

    (k + 1)! > 2 k!


    k!(k+1) > 2 k!

    Here you can see the when k > 1, k!(k+1) > 2 k! and since the original formula is only supposed to work for numbers greater than or equal to 4 its fine.

  7. Jul 11, 2010 #6


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    We are required to show that [tex]n!>2^{n}[/tex], for the k+1 term, multiply by k+1 to obtain:
    Since k+1>2 for k>2 and we are done
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