A) \[\frac{3}{2}\]
B) \[\frac{5}{2}\]
C) \[2\]
D) \[1\]
Correct Answer: C
Solution :
Let \[S={{2}^{1/4}}\cdot {{4}^{1/8}}\cdot {{8}^{1/16}}\cdot {{16}^{1/32}}\] \[={{2}^{1/4}}\cdot {{2}^{2/8}}\cdot {{2}^{3/16}}\cdot {{2}^{4/32}}\] \[={{2}^{\left( \frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{4}{32}+... \right)}}\] Let \[{{S}_{1}}=1+2\cdot \frac{1}{2}+3\cdot \frac{1}{{{2}^{2}}}+4\cdot \frac{1}{{{2}^{3}}}+...\] ? (i) \[\therefore \]\[\frac{1}{2}{{S}_{1}}=\frac{1}{2}+2\cdot \frac{1}{{{2}^{2}}}+3\cdot \frac{1}{{{2}^{3}}}+...\] ? (ii) On subtracting Eq. (i) from Eq. (ii), we get \[\frac{1}{2}{{S}_{1}}=1+\frac{1}{2}+\frac{1}{{{2}^{2}}}+\frac{1}{{{2}^{3}}}+...\] \[=\frac{1}{1-\frac{1}{2}}\] \[\Rightarrow \] \[\frac{1}{2}{{S}_{1}}=2\] \[\Rightarrow \] \[{{S}_{1}}=4\] \[\therefore \] \[S={{2}^{\frac{1}{4}(4)}}=2\]
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