First slide

Geometric progression

Question

Three numbers form an increasing G.P. If the middle number is doubled, then the new numbers are in A.P. The common ratio of the G.P. is

Moderate

A

$2 - \sqrt{3}$
B

$2 + \sqrt{3}$
C

$\sqrt{3} - 2$
D

$3 + \sqrt{2}$

Solution

Let the three numbers be a/r, a, ar. As the numbers form an increasing G.P., so, r > 1. It is given that a/r, 2a, ar are in A.P. Hence,
$\begin{array}{l} 4 a = \frac{a}{r} + ar \\ \Rightarrow r^{2} - 4 r + 1 = 0 \\ \Rightarrow r = 2 \pm \sqrt{3} \\ = 2 + \sqrt{3} [∵ r > 1] \end{array}$

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