In a geometric progression consisting of positive terms, each term equals sum of next two terms. The

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8. Sequences and Series

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In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of its progression is equals

A

$\frac{{\sqrt 5 - 1}}{2}$

B

$\frac{{1 - \sqrt 5 }}{2}$

C

$1$

D

$2\sqrt 5 $

(AIEEE-2007)

Solution

Let the series $a, a r, a r^{2}, \ldots \ldots \ldots$ are in geometric progression

given, $a=a r+a r^{2}$

$\Rightarrow 1=r+r^{2}$

$\Rightarrow r^{2}+r-1=0$

$\Rightarrow r=\frac{-1+\sqrt{1-4 \times-1}}{2}$

$\Rightarrow r=\frac{-1+\sqrt{5}}{2}$

$\Rightarrow r=\frac{\sqrt{5}-1}{2}$

[As terms of $G . P .$ are positive

$\therefore r \text { should be positive }]$

Standard 11

Mathematics

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