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Help me proving this inequality

  1. Nov 21, 2012 #1

    utkarshakash

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    1. The problem statement, all variables and given/known data
    If n is a positive integer, prove that [itex]2^n > 1+n\sqrt{2^{n-1}}[/itex]

    2. Relevant equations

    3. The attempt at a solution
    I am thinking of applying AM GM HM inequality. But which numbers should I take to arrive at this inequality?
     
  2. jcsd
  3. Nov 21, 2012 #2
    ...This may not be correct for any number n that is a positive integer.

    For example, [itex]2^{1}[/itex] is not greater than [itex]1 + n\sqrt{2^{1-1}}[/itex]. In fact, they are equivalent.
     
  4. Nov 22, 2012 #3

    utkarshakash

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    Ok so assume that it is 'greater that or equal to' instead of just 'is greater than' and prove it
     
  5. Nov 22, 2012 #4

    Ray Vickson

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    No: YOU prove it, or at least show some effort towards the solution. Read the Forum rules!

    RGV
     
  6. Nov 22, 2012 #5

    utkarshakash

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    Hey I really don't know how to solve this. I need some hints to get started. I've already stated that I am thinking of solving it using AM GM HM inequality. I know nothing more than this.
     
  7. Nov 28, 2012 #6
    Use mathematical Induction. Do you know what is it ?
     
  8. Nov 28, 2012 #7

    HallsofIvy

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    I think appying the binomial theorem to (1+ 1)n would also work.
     
  9. Nov 28, 2012 #8

    utkarshakash

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    Yes I know but I'm not required to use it. Also I'm not good at it.
     
  10. Nov 28, 2012 #9
    This might be your chance to improve on your technique then. Problems like this, beginning with "Prove for any n...", practically yell out "Use induction on me!" And mostly it's the easiest way to solve them.

    I can see the binomial theorem being used to start 2n = (1+1)n = 1 + n Ʃ..., but getting from that sum to √2n-1 may be more work that induction would be. Maybe I'm overlooking some obvious trick though.
     
  11. Dec 1, 2012 #10
    I understand what you want... But believe instead of using "Arithematic mean ≥ Geometric mean" , its more easier to use mathematical induction anyways...

    But you want to use progression and series only... Ok , well , I throw you off a hint :

    2n =2x2n-1.Now use A.M≥G.M here..
     
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