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: Induction proof of 2^n < n!

  1. Jan 11, 2005 #1
    I posted this elsewhere but realized it ought to be in the homework section. I have to use induction to prove that for n>=4, 2^n < n! is true, but I don't know wehre to start. I have the base case proven, but then I don't know where to go after I have my Inductive Hypothesis that it works for all n's greater than 4. Any help would be very appreciated. Thank you
  2. jcsd
  3. Jan 11, 2005 #2
    NateTG helped me in the original posting, so you can disregard this (unless you want to do it for fun!) Thanks
  4. Jan 11, 2005 #3

    Andrew Mason

    User Avatar
    Science Advisor
    Homework Helper

    Write out the expression for n!

    [tex]n! = n(n-1)(n-2)(n-3)...(n-(n-1))[/tex]

    Then subtract each term by a positive number so that each term in the product is equal to 2.

  5. Jan 11, 2005 #4


    User Avatar
    Homework Helper

    Induction is the easiest way, I think.

    EDIT : Sorry, I just saw the title of your post, and you wanted the induction proof. :smile:

    The initial verification for the case of 4 is easy. Say the inequality holds for some [itex]k[/itex].

    Then [tex]2^{k+1} = 2.2^k < 2.k! < (k+1).k! = (k+1)![/tex] because [itex](k+1) > 2 [/itex] for all [itex]k > 1[/itex].

    And you're done.
    Last edited: Jan 11, 2005
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook