If the first term of an infinite G.P. is 1 and each term is  twice the sum of the succeeding terms, then the common ratio is

Let the G.P. be 1 $\mathrm{r},{\mathrm{r}}^2,{\mathrm{r}}^3,..$

Given, ${\mathrm{t}}_{\mathrm{n}}=2\left({\mathrm{t}}_{\mathrm{n}+1}+{\mathrm{t}}_{\mathrm{n}+2}+{\mathrm{t}}_{\mathrm{n}+3}+\dots \text{ to }\mathrm{\infty }\right)$

$\begin{array}{l}\text{ or }{\mathrm{t}}_{\mathrm{n}}=2\cdot \frac{{\mathrm{t}}_{\mathrm{n}+1}}{1-\mathrm{r}},(\because \text{ common ratio }=\mathrm{r})\\ \therefore =\frac{{\mathrm{t}}_{\mathrm{n}+1}}{{\mathrm{t}}_{\mathrm{n}}};\text{ or }\frac{1-\mathrm{r}}{2}=\mathrm{r}\\ \text{ or }1-\mathrm{r}=2\mathrm{r};\therefore \mathrm{r}=\frac{1}{3}\end{array}$