Question

Three positive numbers form an increasing GP. If the middle term of the GP is doubled, then new numbers are in AP. Then, the common ratio of the GP is
(IIT JEE Main 2014)

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

Answer 1

The correct option is D $2+\sqrt{3}$

Let a, ar, $a{r}^2$ are in GP.
If the middle term of the GP is doubled, then new numbers are in AP.
$\Rightarrow a,2ar,a{r}^2$ are in AP.
$\Rightarrow 4ar-a+a{r}^2$
$\Rightarrow r^2-4r+1=0$
$\Rightarrow r-2\pm \sqrt{3}$
Since the series is an increasing GP so,
$\Rightarrow r-2+\sqrt{3}$