The value of $3^{1 / 4} {.9}^{1 / 8} {.27}^{1 / 16} {.81}^{1 / 32} . . . . . . . . . .,$ upto infinity is:

1

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9

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3

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None of these

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Solution

The correct option is C $3$
$S = 3^{(\frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{4}{32} + . . . . . \infty)}$
This forms a AGP in the power with $a = 1, d = 1, r = \frac{1}{2}$
$S = 3^{\frac{1}{4} (\frac{1}{1} + \frac{2}{2} + \frac{3}{4} + \frac{4}{8} + . . . . . \infty)} S = 3^{\frac{1}{4} ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{1}{1 - \frac{1}{2}} + \frac{1 \times 1 / 2}{{(1 - \frac{1}{2})}^{2}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦} = 3^{\frac{1}{4} (2 + 2)} = 3$
AGP $S_{\infty} = \frac{a}{1 - r} + \frac{d r}{{(1 - r)}^{2}}$

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