Question

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

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

Answer 1

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

Let the three positive numbers be $a,ar,a{r}^2$ which form an increasing G.P.
Also,$a,2ar,a{r}^2$ are in A.P.
$\Rightarrow 2ar-a=a{r}^2-2ar$
$\Rightarrow 4ar=a{r}^2+a$
$\Rightarrow 4r=r^2+1$
$\Rightarrow r^2-4r+1=0$
Using quadratic formula, we get
$r=\frac{4\pm \sqrt{{(-4)}^2-4\left(1\right)\left(1\right)}}{2\left(1\right)}$
$\Rightarrow r=\frac{4\pm \sqrt{12}}{2}$
$\Rightarrow r=\frac{4\pm 2\sqrt{3}}{2}$
$\Rightarrow r=2\pm \sqrt{3}$
But, we know that G.P is increasing
Therefore, $r=2+\sqrt{3}$.