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Prove inequality

  1. Oct 31, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove [tex]\frac{a^{n+1}}{b+c}+\frac{b^{n+1}}{a+c}+\frac{c^{n+1}}{a+b}=(\frac{a^{n}}{b+c}+\frac{b^{n}}{a+c}+\frac{c^{n}}{a+b})*\sqrt[n]{\frac{a^{n}+b^{n}+c^{n}}{3}}[/tex]
    if n>=1 and a,b,c [tex]\in\textsl{R}_{+}[/tex]

    2. Relevant equations



    3. The attempt at a solution
    I tried prove it i some ways but i think any of it don't approach me to solution. I need a clue, don't give me solution.

    PS sorry for my english
     
  2. jcsd
  3. Oct 31, 2009 #2

    Mark44

    Staff: Mentor

    Clue: mathematical induction
     
  4. Oct 31, 2009 #3
    Mathematical Induction is a method of proving a series of mathematical statement labelled by natural numbers
     
  5. Oct 31, 2009 #4

    Mark44

    Staff: Mentor

    Right, and aren't your values of n natural numbers? There is no such restriction on a, b, and c.
     
  6. Oct 31, 2009 #5
    I get

    [tex]a^{2}+b^{2}+c^{2}\geq ab+bc+ac[/tex] for n=1
    is it true?
    what is next step(i never used mathematical induction before)

    if i do it for n=2 it will be proved?
     
    Last edited: Oct 31, 2009
  7. Oct 31, 2009 #6

    Mark44

    Staff: Mentor

    For n = 1 you have to show that
    [tex]\frac{a^{2}}{b+c}+\frac{b^{2}}{a+c}+\frac{c^{2}}{a+b}=(\frac{a^{1}}{b+c}+\frac{b^{1}}{a+c}+\frac{c^{1}}{a+b})*\frac{a^{1}+b^{1}+c^{1}}{3}[/tex]

    It is not sufficient to quit after showing that the original statement is true for n = 2.

    In mathematical induction, you assume that the statement is true for n = k, and use that to show that the statement is also true for n = k + 1.
     
  8. Oct 31, 2009 #7
    Thanks for helping me.
     
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