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

  • question_answer
    In the four numbers first three are in G.P. and last three are in A.P. whose common difference is 6. If the first and last numbers are same, then first will be [IIT 1974]

    A) 2

    B) 4

    C) 6

    D) 8

    Correct Answer: D

    Solution :

    Let the numbers be  \[\frac{a}{r},\ a,\ ar,\ 2ar-a\] ?..(i) Where first three numbers are in G.P. and last three are in A.P. Given that the common difference of A.P. is 6, so \[ar-a=6\] ?..(ii) Also given \[\frac{a}{r}=2ar-a\Rightarrow \frac{a}{r}=2\,(ar-a)+a\] \[\Rightarrow \] \[\frac{a}{r}=2(6)+a,\] from (ii) \[\Rightarrow \] \[\left( \frac{a}{r} \right)-a=12\]\[\Rightarrow \]\[a(1-r)=12r\]\[\Rightarrow \]\[r=-\frac{1}{2}\] From (i) we get, \[a\left[ \left( -\frac{1}{2} \right)-1 \right]=6\]or \[a=-4\] Required numbers from (i) are \[8,\ -4,\ 2,\ 8\] .


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