Question

Three numbers are in G.P. If we double the middle term, then they will be in A.P. The common ratio of the G.P. is

• A. $1\pm \sqrt{5}$
• B. $1\pm \sqrt{3}$
• C. $2\pm \sqrt{3}$
• D. $2\pm \sqrt{5}$

Answer 1

The correct option is C $2\pm \sqrt{3}$

Let the three numbers in G.P. be $a,ar$ and $a{r}^2$. By the given condition, $a,2ar$, and $a{r}^2$ are in A.P. Hence,
$4ar=a+a{r}^2$
$\Rightarrow 4r=1+r^2$
$\Rightarrow r^2-4r+1=0$
$\Rightarrow r=\frac{4\pm \sqrt{16-4}}{2}=2\pm \sqrt{3}$