Results 1 to 3 of 3

Math Help - Use induction

  1. #1
    MHF Contributor alexmahone's Avatar
    Joined
    Oct 2008
    Posts
    1,074
    Thanks
    7

    Use induction

    Use induction to prove that 3^n>n^4 if n\geq8.

    My attempt:

    For n=8, the statement becomes 6561>4096, which is true.

    Assume that the statement is true for n.

    Then, 3^{n+1}>3n^4

    We need to prove that 3n^4>(n+1)^4

    \Leftrightarrow 3n^4>n^4+4n^3+6n^2+4n+1

    \Leftrightarrow -2n^4+4n^3+6n^2+4n+1>0 for n\geq8.

    What's the easiest way of proving this?

    Edit: The above inequality is obviously not true for large n. What should I do, instead?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7

    Re: Use induction

    Quote Originally Posted by alexmahone View Post
    Use induction to prove that 3^n>n^4 if n\geq8.

    My attempt:

    For n=8, the statement becomes 6561>4096, which is true.

    Assume that the statement is true for n.

    Then, 3^{n+1}>3n^4

    We need to prove that 3n^4>(n+1)^4

    \Leftrightarrow 3n^4>n^4+4n^3+6n^2+4n+1

    \Leftrightarrow -2n^4+4n^3+6n^2+4n+1>0 for n\geq8.

    What's the easiest way of proving this?

    Edit: The above inequality is obviously not true for large n. What should I do, instead?
    first of all, your last inequality is wrong. why?

    also, you shouldn't have expanded (n+1)^4. just take the fourth root to get the inequality \sqrt[4]{3} > 1 + \frac{1}{n}, which is true for all n \geq 4 because 1 + \frac{1}{n} \leq 1+ \frac{1}{4} = 1.25 but \sqrt[4]{3} > 1.3.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor alexmahone's Avatar
    Joined
    Oct 2008
    Posts
    1,074
    Thanks
    7

    Re: Use induction

    Quote Originally Posted by NonCommAlg View Post
    first of all, your last inequality is wrong. why?
    Oops, it should have been -2n^4+4n^3+6n^2+4n+1<0.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Strong induction vs. structural induction?
    Posted in the Discrete Math Forum
    Replies: 13
    Last Post: April 21st 2011, 01:36 AM
  2. Induction
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: August 11th 2010, 03:08 AM
  3. Replies: 10
    Last Post: June 29th 2010, 01:10 PM
  4. Mathemtical Induction Proof (Stuck on induction)
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: March 8th 2009, 10:33 PM
  5. induction
    Posted in the Algebra Forum
    Replies: 1
    Last Post: March 19th 2008, 06:12 PM

Search Tags


/mathhelpforum @mathhelpforum