Results 1 to 2 of 2

Math Help - elementary number theory question

  1. #1
    Newbie
    Joined
    Feb 2011
    From
    United States
    Posts
    3

    elementary number theory question

    If x_1, ..., x_n are n positive real numbers such that x_1*x_2* *** *x_n = 1, then prove that x_1 + x_2 + ... + x_n >= n (where >= means greater than or equal to). I'm thinking of using induction, and i know that the cases for n = 1 and n = 2 are true, but I don't know how to proceed from there. Thanks in advance for any help and hints.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485
    Keep in mind that you are actually trying to prove the arithmetic-geometric mean inequality.

    One easy way to do this is to use Lagrange multipliers, optimizing f(x_1,x_2,\ldots,x_n)=x_1+x_2+\ldots+x_n subject to the constraint g(x_1,x_2,\ldots,x_n)=x_1 x_2\cdots x_n=1.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. two elementary number theory problem:
    Posted in the Number Theory Forum
    Replies: 8
    Last Post: October 12th 2010, 06:13 AM
  2. elementary number theory: sum of digits
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: October 17th 2009, 11:40 AM
  3. Elementary Number Theory
    Posted in the Number Theory Forum
    Replies: 7
    Last Post: September 4th 2008, 10:22 PM
  4. Elementary Number Theory - beginnings
    Posted in the Number Theory Forum
    Replies: 10
    Last Post: August 27th 2008, 08:20 PM
  5. elementary Number Theory
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: March 25th 2008, 08:45 PM

Search Tags


/mathhelpforum @mathhelpforum