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 ___.

• 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 the numbers in GP be $a,ar,a{r}^2.$
If the middle term of the GP is doubled, then the new numbers are in AP.
$\Rightarrow a,2ar,a{r}^2$ are in AP.
$\Rightarrow 4ar=a+a{r}^2$
$\Rightarrow r^2-4r+1=0(\because a\text{is positive})$
$\Rightarrow r=2\pm \sqrt{3}$
Since the GP is increasing, we must have
$r=2+\sqrt{3}.$