Question

if the first term of an infinite gp is 1 and each term is twice the sum of succeeding terms then the common ratio id

Answer 1

Dear student
$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{G}.\mathrm{P}\mathrm{be}1,\mathrm{r},{\mathrm{r}}^2,{\mathrm{r}}^3,... \\ \mathrm{Given}:{\mathrm{t}}_{\mathrm{n}}=2\left({\mathrm{t}}_{\mathrm{n}+1}+{\mathrm{t}}_{\mathrm{n}+2}+{\mathrm{t}}_{\mathrm{n}+3}+....\right) \\ \mathrm{or}{\mathrm{t}}_{\mathrm{n}}=2\times \frac{{\mathrm{t}}_{\mathrm{n}+1}}{1-\mathrm{r}}\left[\mathrm{As}\mathrm{common}\mathrm{ratio}=\mathrm{r}\right] \\ \therefore \frac{1-\mathrm{r}}{2}=\frac{{\mathrm{t}}_{\mathrm{n}+1}}{{\mathrm{t}}_{\mathrm{n}}} \\ \mathrm{or}\frac{1-\mathrm{r}}{2}=\mathrm{r} \\ \Rightarrow 1-\mathrm{r}=2\mathrm{r} \\ \Rightarrow 1=3\mathrm{r} \\ \Rightarrow \mathrm{r}=\frac{1}{3} \end{gathered}$$
Regards