First slide

Arithmetico-Geometric Progression

Question

The value of $x = 2^{1 / 4} 4^{1 / 8} 8^{1 / 16} 16^{1 / 32} \dots$ is

Moderate

A

2
B

$3 / 2$
C

1
D

1/2

Solution

$x = 2^{S}$
where $S = \frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{4}{32} + \dots$
$\Rightarrow \frac{1}{2} S = \frac{1}{8} + \frac{2}{16} + \frac{3}{32} + \dots$
Subtracting, we get
$\frac{1}{2} S = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \dots = \frac{1 / 4}{1 - 1 / 2}$
$= \frac{1}{4} \times 2 = \frac{1}{2}$
$\begin{array}{ll} \Rightarrow S = 1 . \\ ∴ x = 2 \end{array}$

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