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State and prove a natural generalization

  1. Sep 22, 2011 #1
    1. The problem statement, all variables and given/known data
    State and prove a natural generalization of "prove that for any three positive real numbers x1, x2, x3, x1/x2 + x2/x3 + x3/x1 [itex]\geq[/itex] 3.


    2. Relevant equations
    AG Inequality is used in subproof (x1/x2 + x2/x3 + x3/x1 [itex]\geq[/itex] 3)


    3. The attempt at a solution
    I don't know what the book means exactly by a "natural generalization". Does it want me to prove the original AG Inequality or relate it somehow to this specific instance of the AG Inequality?
     
  2. jcsd
  3. Sep 22, 2011 #2
    What's special about the statement being true for 3 numbers? Is it true that for any 4 positive real numbers [itex] x_1, x_2, x_3, x_4[/itex]
    [tex]
    \frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_4}+ \frac{x_4}{x_1} \geq 4 \; ?
    [/tex]
    Can you generalize?
     
  4. Sep 22, 2011 #3
    So it's asking for a proof of the form (x1+x2+...+xn)/n [itex]\geq[/itex] [itex]\sqrt[n]{x1x2...xn}[/itex] . So, I should prove the Arithmetic-Geometric Mean Inequality?
     
    Last edited: Sep 22, 2011
  5. Sep 23, 2011 #4
    If by "of the form," you mean for [itex] n [/itex] positive real numbers, then yes. I'm guessing the same trick you used for the [itex] n=3[/itex] case will work in general. If not, proof by induction might work.
     
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