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CEE Kerala Engineering CEE Kerala Engineering Solved Paper-2001

  • question_answer
    The value of n for which the expression\[\frac{{{x}^{n+1}}+{{y}^{n+1}}}{{{x}^{n}}+{{y}^{n}}}\]is arithmetic mean between\[x\]and y, is:

    A)  0                            

    B)         1

    C)  \[-1\]                   

    D)         2

    E)  none of these

    Correct Answer: A

    Solution :

    \[\because \] \[\frac{{{x}^{n+1}}+{{y}^{n+1}}}{{{x}^{n}}+{{y}^{n}}}=\frac{x+y}{2}\] \[\Rightarrow \]               \[2{{x}^{n+1}}+2{{y}^{n+1}}={{x}^{n+1}}+{{x}^{n}}y\]                                                 \[+{{y}^{n+1}}+x{{y}^{n}}\] \[\Rightarrow \]               \[{{x}^{n+1}}-{{x}^{n}}y+{{y}^{n+1}}-x{{y}^{n}}=0\] \[\Rightarrow \]               \[{{x}^{n}}(x-y)-{{y}^{n}}(x-y)=0\] \[\Rightarrow \]               \[(x-y)({{x}^{n}}-{{y}^{n}})=0\] \[\Rightarrow \]               \[{{x}^{n}}={{y}^{n}}\]                                   \[(\because x\ne y)\] \[\Rightarrow \]               \[n=0\]


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