Question

Three positive numbers form an increasing $G.P.$ If the middle term in this $G.P.$ is doubled, then new numbers are in $A.P.$ Then, the common ratio of the $G.P.$ is

• A. $\sqrt 2 + \sqrt 3$
• B. $3 + \sqrt 2$
• C. $2 - \sqrt 3$
• D. $2 + \sqrt 3$

Answer 1

Answer: (D)

$2 + \sqrt 3$

Answer 2

$a, a r, a r^{2} \rightarrow G.P.$
$a, 2 a r, a r^{2} \rightarrow A.P.$
$2 \times 2 a r=a+a r^{2}$
$4 r=1+r^{2}$
$\Rightarrow r^{2}-4 r+1=0$
$r=\frac{4 \pm \sqrt{16-4}}{2}=2 \pm \sqrt{3}$
$r=2+\sqrt{3}$
$r=2-\sqrt{3}$ is rejected
$\because (r>\,1)$
G.P. is increasing.