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JEE Main & Advanced Mathematics Sequence & Series Question Bank Arithmetic Progression

  • question_answer
    If  \[\frac{{{a}^{n+1}}+{{b}^{n+1}}}{{{a}^{n}}+{{b}^{n}}}\] be the A.M. of \[a\] and \[b\], then \[n=\] [MP PET 1995]

    A) 1

    B) \[-1\]

    C)   0

    D)   None of these

    Correct Answer: C

    Solution :

    \[\frac{{{a}^{n+1}}+{{b}^{n+1}}}{{{a}^{n}}+{{b}^{n}}}=\frac{a+b}{2}\] \[\Rightarrow \] \[{{a}^{n+1}}-a{{b}^{n}}+{{b}^{n+1}}-b{{a}^{n}}=0\]\[\Rightarrow \]\[(a-b)({{a}^{n}}-{{b}^{n}})=0\] If\[{{a}^{n}}-{{b}^{n}}=0\]. Then\[{{\left( \frac{a}{b} \right)}^{n}}=1={{\left( \frac{a}{b} \right)}^{0}}\]. Hence\[n=0\].


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