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Proof by induction problem explanation

  1. Feb 18, 2012 #1
    In this link (https://www.physicsforums.com/showthread.php?t=523874 [Broken]) there's an example of proof by induction.

    Somewhere in the explanation there's the phrase:
    Why they decided to multiply both sides by x+1?
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Feb 18, 2012 #2

    eumyang

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    Homework Helper

    It was assumed that
    (1) [itex](1+x)^n\geq 1+nx[/itex]

    And you want to prove that
    (2) [itex](1+x)^{n+1}\geq 1+(n+1)x[/itex]

    By multiplying both sides of (1) by (1+x), you'll get the LHS of the resulting inequality to match the LHS of (2).

    [itex]a^m a = a^m a^1 = a^{m+1}[/itex] by the product property of exponents, so
    [itex](1+x)^n (1+x) = (1+x)^{n+1}[/itex].
     
  4. Feb 18, 2012 #3
    So, the equation will be: [itex](1+x)^{n+1} \geq (1+x)(1+nx)[/itex]?

    From the RHS I can't get 1+(n+1)x, or can I?
     
  5. Feb 18, 2012 #4

    HallsofIvy

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    Well, what do you get when you multiply on the right?
     
  6. Feb 18, 2012 #5
    [itex](1+x)(1+nx)[/itex] is equal to [itex]1+(n+1)x [/itex]?
     
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