Question

Three numbers form an increasing GP. If the middle number doubled, then the new numbers are in AP. The common ratio of the GP is

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

Answer 1

Answer: (A), (B)

The correct options are
A $2-\sqrt{3}$
B $2+\sqrt{3}$

Let the numbers be $a,ar$ and $a{r}^2$ in GP where $r$ is common ratio.

According to the question,

$a,2ar$ and $a{r}^2$ are in AP then

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

$4r=1+r^2$

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

$r=\frac{-(-4)\pm \sqrt{(-4{)}^2-4\times 1\times 1}}{2\times 1}$

$r=\frac{4\pm \sqrt{16-4}}{2}$

$r=\frac{4\pm \sqrt{12}}{2}$

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

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

Hence, this is the value of common ratio of given GP.