Question

If each term of an infinite $\text{G.P.}$ is twice the sum of the terms following it, then the common ratio of the $\text{G.P.}$ is

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

Answer 1

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

Given: Each term of an infinite $\text{G.P.}$ is twice the sum of the terms following it.

To Find: Common ratio

Let $a$ be the first term and $r$ the common ratio of the $\text{G.P.}$

Then,
$a_n=2[a_{n+1}+a_{n+2}+a_{n+3}+\cdots +\infty ]$ for all $n\in N$

$\Rightarrow a{r}^{n-1}=2[a{r}^n+a{r}^{n+1}+a{r}^{n+2}+\cdots +\infty ]$

$\Rightarrow a{r}^{n-1}=\frac{2a{r}^n}{1-r}$

$\Rightarrow 1=\frac{2r}{1-r}$

$\Rightarrow r=\frac{1}{3}$