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JEE Main & Advanced Mathematics Sequence & Series Question Bank Relation between AP., GP. and HP.

  • question_answer
    If \[{{a}^{1/x}}={{b}^{1/y}}={{c}^{1/z}}\]and \[a,\ b,\ c\] are in G.P., then \[x,\ y,\ z\]  will be in       [IIT 1969; UPSEAT 2001]

    A) A.P.

    B) G.P.

    C) H.P.

    D) None of these

    Correct Answer: A

    Solution :

    Let \[{{a}^{1/x}}={{b}^{1/y}}={{c}^{1/z}}=k\Rightarrow a={{k}^{x}},\,b={{k}^{y}},\ c={{k}^{z}}\] Now, \[a,\ b,\ c\]are in G.P. \[\Rightarrow \] \[{{b}^{2}}=ac\Rightarrow {{k}^{2y}}={{k}^{x}}.{{k}^{z}}={{k}^{x+z}}\Rightarrow 2y=x+z\] \[\Rightarrow \] \[x,\ y,\ z\] are in A.P.


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