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JEE Main & Advanced Mathematics Sequence & Series Question Bank Mock Test - Sequences and Series

  • question_answer
    If \[{{a}_{1}},{{a}_{2}},{{a}_{3}}....{{a}_{n}}\]are in H.P. and \[f(k)=\left( \sum\limits_{r=1}^{n}{{{a}_{r}}} \right)-{{a}_{k}}\]then \[\frac{{{a}_{1}}}{f(1)},\frac{{{a}_{2}}}{f(2)},\frac{{{a}_{3}}}{f(3)},...\frac{{{a}_{n}}}{f(n)}\]are in

    A) A.P                  

    B) G.P

    C) H.P                 

    D) none of these

    Correct Answer: C

    Solution :

    [c] \[f(k)+{{a}_{k}}=\sum\limits_{r=1}^{n}{{{a}_{r}}}=\lambda \,(say)\] \[\therefore f(k)=\lambda -{{a}_{k}}\] \[\Rightarrow \frac{{{f}_{(k)}}}{{{a}_{k}}}=\frac{\lambda }{{{a}_{k}}}-1\] \[\therefore \frac{f(1)}{{{a}_{1}}},\frac{f(2)}{{{a}_{2}}},...,\frac{f(n)}{{{a}_{n}}}\] are in A.P. So \[\frac{{{a}_{1}}}{f(1)},\frac{{{a}_{2}}}{f(2)},...\frac{{{a}_{n}}}{f(n)}\]are in H.P.


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