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Proving an equality using induction proof not working

  1. Jan 3, 2016 #1
    1. The problem statement, all variables and given/known data
    I work out the problem completely and it does not equal out. Having problems with two variable induction proofs (n and k) in this problem. Below is as far as I got, jpeg below

    2. Relevant equations


    3. The attempt at a solution
     

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  2. jcsd
  3. Jan 3, 2016 #2
    Our inductive hypothesis is ##\prod\limits_{n=1}^{k}n(2k+2-2n)=2^k(k!)^2##, for some ##k\in\{1,2,...\}##.

    We want to show that ##\prod\limits_{n=1}^{k+1}n(2(k+1)+2-2n)=2^{k+1}((k+1)!)^2##.
     
  4. Jan 3, 2016 #3

    SammyS

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    upload_2016-1-3_22-26-12.png
    That's what you're to prove.

    I think it's clearer if you do the induction step as follows.

    Assume that ##\displaystyle \ \prod_{n=1}^{k}n(2k+2-2n)=2^k(k!)^2 \ ## is true for ##\ k=m\ ## for some ##m>0##. Then show that it's true for ##k = m+1##. You must replace every ##k## with ##m## or ##m+1## as appropriate.

    Note: In the jpeg image that you showed, you needed to have extra parentheses in a number of places.
     
  5. Jan 3, 2016 #4

    Ray Vickson

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    Are you absolutely required to use induction? If not, just writing out the product directly and simplifying is by far the easiest way to do the problem.
     
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