1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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?

  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


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    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

    User Avatar
    Science Advisor
    Homework Helper

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

  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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook