Question

If each term of an infinite G.P. is twice the sum of the terms following it, then the common ratio of such GP. is

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{1}{4}$
• D. $\frac{1}{5}$

Answer 1

The correct option is B $\frac{1}{3}$

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

So, $a=2[ar+a{r}^2+.....]$

$\Rightarrow a=2ar[1+r+r^2+.......]$

$\Rightarrow 1=2r\times \frac{1}{1-r}$ $[\because \text{for a GP,}{S}_{\infty }=\frac{a}{1-r}]$

$\Rightarrow 1-r=2r$

$\Rightarrow 1=3r$

Hence, $r=\frac{1}{3}$