First slide

Geometric progression

Question

In a G.P. consisting of positive terms, each term equals the sum of the next two terms. Then common ratio of the G.P. is

Moderate

A

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

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

$\sqrt{5}$
D

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

Solution

We are given
$a r^{n} = a r^{n + 1} + a r^{n + 2} \forall n \geq 1$
$\Rightarrow 1 = r + r^{2} \Rightarrow r^{2} + r - 1 = 0 \Rightarrow r = \frac{1}{2} (- 1 \pm \sqrt{5})$
As G.P. is of positive terms, $r > 0 .$ Thus, $r = \frac{1}{2} (\sqrt{5} - 1)$

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