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Proof by induction

  1. Feb 15, 2012 #1
    I am confused by what the book is saying, can someone explain how they got the thing I circled in red?

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  2. jcsd
  3. Feb 15, 2012 #2
    your book explains it on the side, it is using the inductive hypothesis and a fact about integers to say that (2^k+1)+2 < 2^k+2^k=2(2^k)=2^(k+1)
     
  4. Feb 15, 2012 #3

    SammyS

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    What don't you understand? The thing circled in red is explained just to the right of that.

    You're assuming that [itex]2k+1<2^k[/itex] .

    It's also true that: [itex]\text{ for }\ k ≥ 2\,,\ \ 2 < 2^k[/itex]

    Therefore, [itex]2k+1+2<2^k+2<2^k+2^k\,. [/itex]
     
  5. Feb 15, 2012 #4

    Ray Vickson

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    Their comments in blue tell you exactly how they got it.

    RGV
     
  6. Feb 15, 2012 #5
    How did they decide 2 < 2k? Where did the 2 come from
     
  7. Feb 15, 2012 #6
    they expanded 2(k+1), which is the thing you're trying to prove the relationship about, into (2k+1)+2 on the LHS, and then on the RHS they're just trying to make relatable to 2k+1 so they used the relationship 2 < 2k to facilitate it.
     
  8. Feb 15, 2012 #7
    Oh I think I see it now. So when they use 2 < 2k its kinda like stating 2 = 2k so (2k+1) + 2 < 2k + 2k
     
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