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Proving inequality by induction,given a condition

  1. Nov 4, 2008 #1
    1. The problem statement, all variables and given/known data
    If [tex]x_{1} x_{2} \cdots x_{n}=1[/tex] (1)
    show that
    [tex]x_{1}+x_{2}+\cdots+x_{n} \geq n[/tex] (2)


    3. The attempt at a solution

    I attempted as follows. I started with

    [tex]x_{1} + \frac{1}{x_{1}} \geq 2[/tex] , which is an inequality I already know how to prove.

    Then using Eq.(1) I get
    [tex]x_{1} + x_{2} x_{3} \cdots x_{n} \geq 2[/tex]

    Continuing from this point , for example started from another point [tex]x_{2}[/tex] and repeating the procedure for all [tex]n[/tex] , I get no where. I cannot think of another path to take.

    If i try to do it by induction, I cannot assume that the equation holds for [tex]n[/tex] numbers , and try to prove for [tex]n+1[/tex] numbers, as by including [tex]x_{n+1}[/tex], Eq.(1) and Eq.(2) need not hold anymore but
    [tex]x_{1} x_{2} \cdots x_{n} x_{n+1}=1[/tex]
    [tex]x_{1}+x_{2}+\cdots+x_{n} +x_{n+1}\geq n +1[/tex]

    Edit:
    Assuming all x's are nonnegative
     
    Last edited: Nov 4, 2008
  2. jcsd
  3. Nov 4, 2008 #2

    morphism

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    I'm assuming all the x_i are nonnegative real numbers.

    Do you really want to do this by induction? It follows pretty easily from the AM-GM inequality.
     
  4. Nov 4, 2008 #3

    Dick

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    Alternatively, you can use a Lagrange multiplier to extremize the sum of the xi's subject to the constraint that their product is 1.
     
  5. Nov 4, 2008 #4
    Well I could use AM-GM , or the method by langrange multiplier but I am interested in how to apply the principle of induction itself when such a condition is given, or if its possible at all .
     
  6. Nov 4, 2008 #5

    Dick

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    I don't really see how to get at this with induction.
     
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