Results 1 to 4 of 4

Math Help - Product of n numbers = 1, sum >= n - Proof

  1. #1
    Newbie
    Joined
    Jul 2009
    Posts
    23

    Product of n numbers = 1, sum >= n - Proof

    Hi everyone,

    I have a little problem with the following exercise.

    I have to prove by mathematical induction that for all n real numbers, which’s product is 1, their sum is greater than or equal to n.

    I’d be very happy if someone could help me with this.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Jan 2009
    Posts
    108
    Quote Originally Posted by thomasdotnet View Post
    Hi everyone,

    I have a little problem with the following exercise.

    I have to prove by mathematical induction that for all n real numbers, which’s product is 1, their sum is greater than or equal to n.

    I’d be very happy if someone could help me with this.
    Let n be 2. We'll use -2 and -0.5 as our real numbers. The product is 1. Their sum is -2.5 which is not greater than or equal to n.

    Are you sure you copied the problem correctly? Does it say positive real numbers?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Thanks
    1
    Hello thomasdotnet.

    Welcome to Math Help Forum!
    Quote Originally Posted by thomasdotnet View Post
    Hi everyone,

    I have a little problem with the following exercise.

    I have to prove by mathematical induction that for all n real numbers, which’s product is 1, their sum is greater than or equal to n.

    I’d be very happy if someone could help me with this.
    Assuming that the numbers are all non-negative, the easiest proof is to use the AM-GM inequality: that the arithmetic mean of a set of non-negative real numbers is greater than or equal to their geometric mean.

    If the product of the n numbers is 1, then their GM is 1. Hence their AM \ge 1 \Rightarrow their sum \ge n.

    The proof by Induction of this inequality can be found here: Inequality of arithmetic and geometric means - Wikipedia, the free encyclopedia

    Grandad
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Jul 2009
    Posts
    23
    Actually I did copy the question correctly, but what you said wytiaz makes sense, so think that it's only about positive real numbers. Thanks.

    Thank you for pointing me to the AM-GM inequality Grandad, that's exactly what I needed.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. probability on product of numbers
    Posted in the Statistics Forum
    Replies: 1
    Last Post: January 7th 2011, 08:31 AM
  2. Product of Complex Numbers proof
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: December 8th 2010, 02:31 PM
  3. Product of complex numbers.
    Posted in the Algebra Forum
    Replies: 4
    Last Post: September 12th 2010, 10:09 AM
  4. Replies: 5
    Last Post: June 30th 2010, 01:48 PM
  5. Replies: 5
    Last Post: April 3rd 2010, 01:17 PM

Search Tags


/mathhelpforum @mathhelpforum