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:

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:

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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