Question

If each term of an infinite G.P. is twice the sum of the terms following it, then the common ratio of G.P. 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 $a$ be the first term and $r$ the common ratio of the G.P.
Then,
$a_n=2[a_{n+1}+a_{n+2}+a_{n+3}+\cdots ]$, for all $n\in N$ [Given]
$\Rightarrow a{r}^{n-1}=2[a{r}^n+a{r}^{n+1}+\cdots ]$
$\Rightarrow a{r}^{n-1}=\frac{2a{r}^n}{1-r}$
$\Rightarrow 1=\frac{2r}{1-r}$
$\Rightarrow r=\frac{1}{3}$