Question

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the 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 G.P.
Now, $a,2ar,a{r}^2$ are in A.P.
So, $2ar=\frac{a+a{r}^2}{2}$
$\Rightarrow r^2-4r+1=0$
$\Rightarrow r=\frac{4\pm \sqrt{16-4}}{2}=2\pm \sqrt{3}$
As, G.P. is increasing so,
$r=2+\sqrt{3}$