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Proof by induction question

  1. Sep 15, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove that 3^n >= 2n+1 for all natural numbers.


    2. Relevant equations
    3^n >= 2n+1 [is bigger or equal to]


    3. The attempt at a solution

    3*1>=2+1
    True for n=1

    Assumption: 3^k>=2k+1

    3^(k+1)>=2k+3
    3^k*3>=2k+3
    (2k+1)*3>=2k+3 <---can I just substitute 2k+1 into 3^k as per my assumption, because 3^k is bigger, and (2k+1)*3 is bigger than 2k+3? If so, is the proof complete?

    Thanks in advance.
     
  2. jcsd
  3. Sep 15, 2012 #2

    Mark44

    Staff: Mentor

    No, you can't just assert this -- you have to show it.

    Use the fact that 3^(k + 1) = 3 * 3k, and use your assumption that 3^i >= 2k + 1.
     
  4. Sep 15, 2012 #3
    Sorry, still at a loss.
     
  5. Sep 16, 2012 #4

    Mark44

    Staff: Mentor

    I already provided a strategy for you.
    Did you not understand?
     
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