Question

Three positive numbers form an increasing GP. If the middle term in this GP is doubled, then new numbers are in AP. Then, the common ratio of the GP is

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

Answer 1

The correct option is A

$2+\sqrt{3}$

Explanation for the correct answer:

Let the three terms of the geometric progression be $\frac{a}{r},a,ar$

It is given that if the middle term is doubled then the resultant terms will form an arithmetic progression

$\Rightarrow$ $\frac{a}{r},2a,ar$ are in an arithmetic progression

$\Rightarrow$ $2a=\frac{{\displaystyle \frac{a}{r}}+ar}{2}$

$\Rightarrow$ $4a=\frac{a{r}^2+a}{r}$

$\Rightarrow$ $4ar=a{r}^2+a$

$\Rightarrow$ $r^2-4r+1=0$

Using formula for quadratic equation we get,

$r=\frac{4\pm \sqrt{4^2-4}}{2}$

$\Rightarrow$ $r=\frac{4\pm 2\sqrt{3}}{2}$

$\Rightarrow$ $r=2\pm \sqrt{3}$

As it is an increasing geometric progression $r$ cannot be less than $1$

Hence, $r=2+\sqrt{3}$

Hence, the common ratio of the geometric progression is $2+\sqrt{3}$

Hence, option A is the correct answer.