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

  • question_answer
    The sum of an infinite GP is x and the common ratio r is such that \[\left| r \right|<1\]. If the first term of the GP is 2, then which one of the following is correct?

    A) \[-1<x<1\]

    B) \[-\infty <x<1\]

    C) \[1<x<\infty \]

    D) None of these

    Correct Answer: C

    Solution :

    [c] GP = x \[\frac{a}{1-r}=x\] (where, a = 1st term and r = common ratio) \[\Rightarrow \,\,\frac{2}{1-r}=x\] ... (i)  (\[\because \]  Given \[a=2\] and \[|r|<1\]) \[\Rightarrow \,\,\,-1<r<1\Rightarrow 1>-r>-1\] \[\Rightarrow \,\,1+1>1-r>1-1\] \[\Rightarrow \,\,\,0<1-r<2\] \[\Rightarrow \,\,\,\frac{1}{1-r}>\frac{1}{2},\frac{2}{1-r}>1\] from equation (i) \[x>1\] Hence, \[1<x<\infty .\]


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