Question

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

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

Answer 1

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

Let the three numbers be $\frac{a}{r},a,ar$. As the numbers form an increasing G.P., so, $r>1$. It is given that $\frac{a}{r},2a,ar$ are in A.P.
Hence,
$4a=\frac{a}{r}+ar$
$\Rightarrow r^2-4r+1=0$
$\Rightarrow r=2\pm \sqrt{3}$
$=2+\sqrt{3}[\because r>1]$