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Induction problem

  1. Aug 26, 2005 #1
    Hello, I have this problem in my calculus class where I have to prove a formula with induction.

    the problem is:

    ( 1 * 3 * 5 * .... * (2n - 1) ) / ( 1 * 2 * 3 * ... * n) =< 2^n

    =< = equal or lesser than

    P(1) is easy to solve, and so is P(k), but I start having problems with P(k+1) to prove the formula.. can someone give me a hand? =)

    thanks!

    /Maximilian
     
  2. jcsd
  3. Aug 26, 2005 #2

    HallsofIvy

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    Yes, P(1) is "easy to solve": 1/1= 1< 21.
    I'm not sure what you mean by saying P(k) is "easy to solve"- there's nothing to solve there!

    You do, of course, assume that (1*2*...*(2k-1))/(1*2*...*k)<2k

    Now, for P(k+1), you have to look at (1*2*..*(2k-1)(2(k+1)-1)/(1*2*...*k*(k+1))
    2(k+1)-1= 2k+1 of course, so this is
    (1*2*...*(2k-1)*(2k+1)/(1*2*...*k*(k+1)= {(1*2*...*(2k-1))/(1*2*...*k)}{(2k+1)/(k+1)}< 2k{(2k+1)/(k+1)}.

    Looks to me like you need to prove that (2k+1)/(k+1)< 2 for all k! I would be inclined to write that as a "lemma" first and use induction to prove it.
     
  4. Aug 26, 2005 #3
    (2n-1)!/{(2^(n-1)(n-1)!(n!)}=<2^n

    {2^n(n!)}{2^(n-1)(n-1)!}>=(2n-1)!

    (2n-1)!/(n!)(n-1)!=<2^(2n-1)

    C(2n-1,n)=<2^(2n-1)
    True?
     
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