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Math Help - progressions!

  1. #1
    Member grgrsanjay's Avatar
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    progressions!

    if a,b,c,d and p are distinct real numbers such that
    (a^2 +b^2 +c^2 )p^2 -2(ab + bc + cd)p+(b^2 + c^2 + d^2) is less than or equal to zero..

    then prove that a,b,c,d are in geometric progression.




    thanks for you reply.
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  2. #2
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    Opalg's Avatar
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    Quote Originally Posted by grgrsanjay View Post
    if a,b,c,d and p are distinct real numbers such that
    (a^2 +b^2 +c^2 )p^2 -2(ab + bc + cd)p+(b^2 + c^2 + d^2) is less than or equal to zero.

    then prove that a,b,c,d are in geometric progression.
    If a quadratic (in p), in which the coefficient of p^2 is non-negative, can take a value less than or equal to zero, then its discriminant must be greater than or equal to zero. Therefore (ab+bc+cd)^2 - (a^2+b^2+c^2)(b^2+c^2+d^2) \geqslant0. Multiply that out and rearrange it as -(b^2-ac)^2 -(c^2-bd)^2 - (ad-bc)^2 \geqslant0. Can you take it from there?
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  3. #3
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    a:b:c:d ; p
    -2:-4:-8:-16 ; 2
    2:-4:8:-16 ; -2

    For your info, that pattern exists.
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  4. #4
    Member grgrsanjay's Avatar
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    thanks! i got it
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