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8. Sequences and Series
hard
गुणनफल $2^{\frac{1}{4}} \cdot 4^{\frac{1}{16}} \cdot 8^{\frac{1}{48}} \cdot 16^{\frac{1}{128}}$ $\infty$ तक बराबर है
A
$2^{\frac{1}{2}}$
B
$2^{\frac{1}{4}}$
C
$2$
D
$1$
(JEE MAIN-2020)
Solution
$2^{\frac{1}{4}} \cdot 4^{\frac{1}{16}} \cdot 8^{\frac{1}{48}} \cdot 16^{\frac{1}{128}} \cdot \ldots . \infty$
$=2^{\frac{1}{4}} \cdot 2^{\frac{2}{16}} \cdot 2^{\frac{3}{48}} \cdot 2^{\frac{4}{128}} \cdot \ldots \infty$
$=2^{\frac{1}{4}} \cdot 2^{\frac{1}{8}} \cdot 2^{\frac{1}{16}} \cdot 2^{\frac{1}{32}} \cdot \ldots \infty$
$=2^{\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\ldots \infty}{32}=(2)\left(\frac{1 / 4}{1-1 / 2}\right)=2^{1 / 2}$
Standard 11
Mathematics
