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Q.

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

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a

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

b

$\sqrt{5}$

c

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

d

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

answer is D.

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Detailed Solution

$t_{1} = t_{2} + t_{3}$

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

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

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

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

$∴ r = \frac{- 1 + \sqrt{5}}{2} (∴ r > 0)$

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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 this progressions equals