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

  • question_answer
    If \[{{G}_{1}}\] and \[{{G}_{2}}\] are two geometric means and A the arithmetic mean inserted between two numbers, then the value of \[\frac{G_{1}^{2}}{{{G}_{2}}}+\frac{G_{2}^{2}}{{{G}_{1}}}\]is [DCE 1999]

    A) \[\frac{A}{2}\]

    B) A

    C) 2 A

    D) None of these

    Correct Answer: C

    Solution :

    Let the two numbers be p and q. \[\therefore \,\,{{G}_{1}}={{p}^{2/3}}\,\,{{q}^{1/3}},\,\,{{G}_{2}}={{p}^{1/3}}\,\,\,{{q}^{2/3}}\] \[\therefore \,\frac{G_{1}^{2}}{{{G}_{2}}}+\frac{G_{2}^{2}}{{{G}_{1}}}=\frac{{{p}^{4/3}}\,\,{{q}^{2/3}}}{{{p}^{1/3}}\,\,{{q}^{2/3}}}+\frac{{{p}^{2/3}}\,{{q}^{4/3}}}{{{p}^{2/3}}\,{{q}^{1/3}}}\] \[=p+q=2\times \,\left( \frac{p+q}{2} \right)\,=2A\].


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