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

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

Let $a,ar,a{r}^2$ are in GP, where (r > 1)
On multiplying middle term by $2$, we have
$a,2ar,a{r}^2$ are in an AP
$\Rightarrow 4ar=a+a{r}^2$
$\Rightarrow r^2-4r+1=0$
$\Rightarrow r=\frac{4\pm \sqrt{16-4}}{2}$
$=2\pm \sqrt{3}$
$\Rightarrow r=2+\sqrt{3}$ [since, AP is increasing]