Question

Three positive numbers from an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G.P. is:

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

Answer 1

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

Let $a,ar,a{r}^2$ are in G.P.
According to the question
a, 2ar, $a{r}^2$ are in A.P.
$\Rightarrow 2\times 2ar=a+a{r}^2$
$\Rightarrow 4r=1+r^2\Rightarrow r^2-4r+1=0$
$r=\frac{4\pm \sqrt{16-4}}{2}=2\pm \sqrt{3}$
Since r > 1
$\therefore r=2-\sqrt{3}$ rejected
Hence, $r=2+\sqrt{3}$