Results 1 to 2 of 2

Math Help - Prove the following statement by induction: For all n ≥ 4, n < 3ⁿ

  1. #1
    Junior Member
    Joined
    Jan 2010
    Posts
    50

    Prove the following statement by induction: For all n ≥ 4, n < 3ⁿ

    I'm probably just being foolish, but I'm having trouble proving it true for n+1, based on the assumption that it's true for n.

    Thanks for your help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Quote Originally Posted by feyomi View Post
    I'm probably just being foolish, but I'm having trouble proving it true for n+1, based on the assumption that it's true for n.

    Thanks for your help.
    Hi feyomi,

    P(k)

    k^3\ <\ 3^k

    P(k+1)

    (k+1)^3\ <\ 3^{k+1}

    Proof

    Examine P(k+1) to see if P(k) being true will cause P(k+1) to be true

    (k+1)^3=k^3+3k^2+3k+1

    3^{k+1}=(3)3^k=3^k+3^k+3^k

    Hence, if k^3\ <\ 3^k

    we ask if 3k^2+3k+1\ <\ 3^k+3^k

    If k\ \ge\ 4,\ 3k^2\ <\ 4k^2\ \Rightarrow\ 3k^2\ <\ k^3

    hence we ask if 3k+1\ <\ 3^k

    3k+1\ <\ (k)k^2\ ?

    3k\ <\ k^2\ for\ k\ \ge\ 4

    hence 3k+1\ <\ k^3,\ for\ k\ \ge\ 4
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. how to prove a closed set in Rⁿ
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 23rd 2011, 11:50 PM
  2. Proving a statement by Induction
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: April 20th 2011, 02:57 AM
  3. Replies: 1
    Last Post: January 22nd 2011, 04:14 AM
  4. Prove the probability statement
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: February 9th 2010, 03:27 PM
  5. Replies: 2
    Last Post: October 1st 2009, 11:58 AM

Search Tags


/mathhelpforum @mathhelpforum