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The first term of an infinite GP. is 1 and each term is twice the sum of the succeeding terms. then the sum of the series is.

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$\frac{5}{2}$

$\frac{3}{2}$

answer is C.

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Detailed Solution

Let the GP .be $1, r, r^{2}, \dots \dots \dots \infty$

Given $x_{n} = 2 (x_{n + 1} + x_{n + 2} + \dots . to \infty)$

$∴ x_{n} = 2 \frac{x_{n + 1}}{1 - r} ∴ \frac{x_{n + 1}}{x_{n}} = \frac{1 - r}{2} \Rightarrow r = \frac{1 - r}{2} ∴ r = \frac{1}{3} 1 + \frac{1}{3} + {(\frac{1}{3})}^{2} + \dots . \infty = \frac{1}{1 - \frac{1}{3}} = \frac{3}{2}$

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