KVPY Sample Paper KVPY Stream-SX Model Paper-22

  • question_answer
    If \[x={}^{n}{{C}_{n\,-1}}+{}^{n+1}{{C}_{n\,-1}}+.....+{}^{2n\,-1}{{C}_{n\,-1}}\] then \[\frac{x+1}{n+1}\] is

    A) an integer iff n is odd integer

    B) an integer iff n is an even integer

    C) never integer

    D) always integer

    Correct Answer: D

    Solution :

    \[\therefore \]\[\frac{x+1}{n+1}=\frac{^{2n}{{C}_{n}}}{n+1}=\frac{^{2n\,\,+\,1}{{C}_{n\,+\,1}}}{2n+1}\in I\] if
    \[2n+1\] & \[n+1\] are co-prime
                Let        \[2n+1=\lambda \,{{I}_{1}}\] & \[n+1=\lambda \,{{I}_{2}}\]
    \[\therefore \]\[\frac{2n+1}{\lambda }-\frac{2\,(n+1)}{\lambda }={{I}_{1}}-2{{I}_{2}}\]
    \[-\frac{1}{\lambda }={{I}_{1}}-2{{I}_{2}}\in I\]
    \[\therefore \]      \[\lambda =\pm \,1\]
    So \[2n+1\] & \[n+1\] are co-prime
    \[\therefore \]      \[\frac{x+1}{n+1}\] is always integer

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