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  4. If (1/x)+(1/2 x2)+(1/4 x3)+(1/8 x4)+(1/16 x5)+ ldots ldots . . ∞=(1/64), then the value of x is

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Q. If $\frac{1}{x}+\frac{1}{2 x^2}+\frac{1}{4 x^3}+\frac{1}{8 x^4}+\frac{1}{16 x^5}+\ldots \ldots . . \infty=\frac{1}{64}$, then the value of $x$ is

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Solution:

$\frac{\frac{1}{x}}{1-\frac{1}{2 x}}=\frac{1}{64} \Rightarrow \frac{\frac{1}{x}}{\frac{2 x-1}{2 x}}=\frac{1}{64}$
$\frac{2}{2 x-1}=\frac{1}{64} \Rightarrow 128=2 x-1 \quad \Rightarrow \quad x=\frac{129}{2}$