Question

Three numbers form an increasing G .P . If the middle number is doubled, then the new numbers are in A.P. The common ratio of G .P. is-

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

Answer 1

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

Let the three numbers in G.P. be $a,ar,a{r}^2$.
Also, by the given conditions the new numbers $a,2ar,a{r}^2$ are in A.P.
Therefore, $4ar=a+a{r}^2$
$\Rightarrow 4r=1+r^2$
$\Rightarrow r^2-4r+1=0$
$\Rightarrow r=\frac{4\pm \sqrt{16-4}}{2}$
$\Rightarrow r=\frac{4}{2}\pm \frac{\sqrt{12}}{2}$
$\Rightarrow r=2\pm \frac{2\sqrt{3}}{2}$
$\Rightarrow r=2\pm \sqrt{3}$
For increasing G.P, $r=2+\sqrt{3}$
Hence, the correct option is 'B'.