Question

Three numbers from an increasing are in $G.P$. If the middle term is doubled, then the new number are in $A.P$. Find the common ratio of the $G.P$.

Answer 1

Let the No. be a, ar, $a{r}^2$

According to question,

a, $2ra,a{r}^2$ are in AP

$b-a=c-b$

$2b=a+c$

$2\left(2ar\right)=a+a{r}^2$

$4r=1+r^2$

$r^2\pm 4r+1=0$

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

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