Results 1 to 6 of 6

Math Help - Analysis Proof

  1. #1
    Junior Member
    Joined
    Jul 2006
    Posts
    43

    Analysis Proof

    How would you go about proving this:

    Let c be greater than 1. Show that c^n is greater than or equal to c for all n greater than or equal to 1 and c^n is greater than c for all n greater than 1. Further than c^m is greater than c^n for all m greater than n.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,963
    Thanks
    1784
    Awards
    1
    If c>1 then c>0 therefore multiply by c and get c^2>c>1.
    Now you can do it by induction.
    If c^k>c^(k-1) then c^(k+1)>c^k; we multiply by c.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    10,211
    Thanks
    419
    Awards
    1
    Quote Originally Posted by JaysFan31 View Post
    How would you go about proving this:

    Let c be greater than 1. Show that c^n is greater than or equal to c for all n greater than or equal to 1 and c^n is greater than c for all n greater than 1. Further than c^m is greater than c^n for all m greater than n.
    Quote Originally Posted by Plato View Post
    If c>1 then c>0 therefore multiply by c and get c^2>c>1.
    Now you can do it by induction.
    If c^k>c^(k-1) then c^(k+1)>c^k; we multiply by c.
    Plato, you are probably correct. However the original problem statement does not specify that m, n are positive integers. (Though I suspect it's an oversight in the statement of the problem.)

    -Dan
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,963
    Thanks
    1784
    Awards
    1
    Oh come Quark!
    In the first post it was stated “for all n greater than or equal to 1”.
    In fact, I only addressed the first of his/her questions.
    However, I think it is logical to assume that in the second question that both n & m are positive integers. But again I did not give an answer to the second question.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by JaysFan31 View Post
    Further than c^m is greater than c^n for all m greater than n.
    The exponent function is defined as,
    a^n=exp( n ln a) for a>0.
    This since,
    exp (x) is an increasing function (look at is derivative sign) it follows that if,
    n>m
    then,
    n ln a > m ln a , since a>1 thus, ln a>0
    Thus,
    exp(n ln a)> exp( m ln a) using the increasing function theorem
    Thus,
    a^n>a^m
    Q.E.D.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    10,211
    Thanks
    419
    Awards
    1
    Quote Originally Posted by Plato View Post
    Oh come Quark!
    In the first post it was stated “for all n greater than or equal to 1”.
    In fact, I only addressed the first of his/her questions.
    (Shrugs) There is nothing in the statement of even the first part of the problem that states that n is an integer either. Yes, I was generalizing, but my comment still applies.

    Quote Originally Posted by Plato View Post
    However, I think it is logical to assume that in the second question that both n & m are positive integers. But again I did not give an answer to the second question.
    I believe I was clear in stating that I also had assumed the original question was intended to have m, n integral. However, as the problem statement (both statements in fact) are true for real m, n I didn't feel comfortable in making the assumption that the problem had to intend this.

    Just making sure all bases are covered. (Uh, no pun intended there!)

    -Dan
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Analysis proof
    Posted in the Differential Geometry Forum
    Replies: 11
    Last Post: April 29th 2010, 09:49 AM
  2. analysis of a proof
    Posted in the Discrete Math Forum
    Replies: 17
    Last Post: November 21st 2009, 07:59 AM
  3. analysis of proof
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 16th 2009, 06:04 AM
  4. Proof analysis
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 9th 2008, 03:20 AM
  5. Analysis Proof
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 25th 2006, 02:53 PM

Search Tags


/mathhelpforum @mathhelpforum