First slide

Geometric progression

Question

Three positive numbers form an increas-ing G.P. If the middle term in this G.P. is tripled, the new numbers are in A.P. Then the common ratio of G.P.is:

Moderate

A

$3 + 2 \sqrt{2}$
B

$2 \sqrt{2} - \sqrt{3}$
C

$2 + 2 \sqrt{3}$
D

$2 \sqrt{2} + \sqrt{3}$

Solution

Let the three positive numbers in the increas-ing G.P. be
$\frac{a}{r}, a, a r$ where $r > 1$
We are given, $\frac{a}{r}, 3 a, a r$ are in A.P.
$\begin{array}{l} ∴ \frac{a}{r} + a r = 2 (3 a) \\ \Rightarrow r^{2} - 6 r + 1 = 0 \\ \Rightarrow r = 3 \pm 2 \sqrt{2} \end{array}$
As $r > 1, r = 3 + 2 \sqrt{2}$

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