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

Answer 1

Let the numbers be $a,ar,a{r}^2$

If $ar$ is doubled then,

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

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

$\Rightarrow 4r=r^2+1$ by dividing both sides by $a$

$\Rightarrow r^2-4r+1=0$ which is quadratic in $r$

$\Rightarrow {\left(r\right)}^2-2\times r\times 2+2^2-2^2+1=0$ by using the completing the square method

$\Rightarrow {(r-2)}^2-4+1=0$

$\Rightarrow {(r-2)}^2=3$

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

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

As it is an increasing G.P we have $r=\sqrt{3}+2$