The product $2^{\frac{1}{4}} \cdot 4^{\frac{1}{16}} \cdot 8^{\frac{1}{48}} \cdot 16^{\frac{1}{128}} \cdot \ldots .$ to $\infty$ is equal to

  • [JEE MAIN 2020]
  • A

    $2^{\frac{1}{2}}$

  • B

    $2^{\frac{1}{4}}$

  • C

    $2$

  • D

    $1$

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Let ${a_n}$ be the ${n^{th}}$ term of the G.P. of positive numbers. Let $\sum\limits_{n = 1}^{100} {{a_{2n}}} = \alpha $ and $\sum\limits_{n = 1}^{100} {{a_{2n - 1}}} = \beta $, such that $\alpha \ne \beta $,then the common ratio is

  • [IIT 1992]