1. Not finding help here? Sign up for a free 30min 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!

How to prove this ?

  1. Jul 27, 2006 #1
    How to prove that [tex]2^n > n^2[/tex] when n>4 ?
     
  2. jcsd
  3. Jul 27, 2006 #2

    0rthodontist

    User Avatar
    Science Advisor

    You could use the fact that both functions are continuous and never cross after that point. If you are only interested in integer n then you could use induction.
     
  4. Jul 27, 2006 #3
    *Waves hand*
    An exponential function grows faster than a polynomial.
     
  5. Jul 27, 2006 #4

    0rthodontist

    User Avatar
    Science Advisor

    That is not sufficient--it does not rule out the possibility of some point before infinity but after 4 where the inequality does not hold.

    For integer n, induction is the way to go. Otherwise, you can show directly that the second derivative of 2^n is greater than that of n^2 for n > 4, and the first derivative of 2^n is greater than that of n^2 for n = 4. From this it is possible to infer the conclusion by integration.
     
  6. Jul 28, 2006 #5
    If I don't want to use induction but want to prove it mathematically ,how to do it ?
     
  7. Jul 28, 2006 #6

    VietDao29

    User Avatar
    Homework Helper

    ???
    Huh?
    What do you mean by mathematically? What's wrong with using induction, by the way? As a matter of fact, it's completely valid!
     
  8. Jul 28, 2006 #7

    0rthodontist

    User Avatar
    Science Advisor

    Induction would be simpler for this case if you want to prove it for integers. The other way I mentioned is using the function [tex]f(x) = 2^x - x^2[/tex]. You should be able to show that [tex]f'(4) > 0[/tex] and also that [tex]f''(x) > 0[/tex] for any x larger than 4. And you know that f(4) = 0. Then use
    [tex]\int_a^t g'(x) dx + g(a) = g(t)[/tex]
    to show that [tex]f'(x) > 0[/tex] for x > 4 and then that [tex]f(x) > 0[/tex] for x > 4.
     
    Last edited: Jul 28, 2006
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: How to prove this ?
  1. How to prove this? (Replies: 8)

  2. How to prove that? (Replies: 2)

  3. How to prove it (Replies: 3)

  4. How to prove this? (Replies: 7)

  5. How to prove this? (Replies: 6)

Loading...