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A,b,c satisfying given conditions

  1. Apr 22, 2014 #1

    utkarshakash

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    1. The problem statement, all variables and given/known data
    Let a,b,c be distinct real numbers satisfying [itex]a^3+b^3+6abc = 8c^3 [/itex] then which of the following may be correct?

    A) a,c,b are in Arithmetic Progression
    B) a,c,b are in Harmonic Progression
    C) a+bω-2cω^2 =0
    D) a+bω^2-2cω=0


    3. The attempt at a solution

    I could prove the first statement but I really can't figure out how to deal with the other options. Since the first statement is correct all I know is that 2c=a+b. Suppose I replace a with 2c-b in the third statement and simplify it, I get 4c-b+ω(2c+b) = 0. But I guess LHS can't be zero(I'm not sure about this).
     
  2. jcsd
  3. Apr 22, 2014 #2
    The following should help:
    $$x^3+y^3+z^3-3xyz=(x+y+z)(x+\omega y+\omega^2 z)(x+\omega^2 y+\omega z)$$
    Can you figure out what ##x,y,z## are for the given problem? :)
     
  4. Apr 22, 2014 #3

    AlephZero

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    For B), do you remember an inequality between the arithmetic and geometric means of a set of numbers?
     
  5. Apr 22, 2014 #4
    I am not sure but OP already proved that a,b,c are in AP. Why check the B case now? :/
     
  6. Apr 23, 2014 #5

    AlephZero

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    The question asks if a b and c may be in AP, GP, etc.

    The OP proved they may be in AP. There could be other values that are in GP.

    In fact they could be in a GP, except that the question says they are distinct real numbers.

    NOTE: I'm can't remember what I was thinking about when I posted #3 - ignore it!

    You can check for a GP by substituting ##b = ka##, and ##c = k^2a##.
     
  7. Apr 23, 2014 #6
    It does state that they are distinct. :)
     
  8. Apr 23, 2014 #7

    AlephZero

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    You replied to my post #5 before I had finished editing the typos :smile:
     
  9. Apr 23, 2014 #8
    :biggrin:

    So no need to check the other cases now. :)
     
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