The sum of first two terms of an infinite geometric series is 15 and each term of the series is equal to the sum of all the terms following it. Find the series ?

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Please find below the solution to the asked query:

$$\begin{gathered} 38. \\ \mathrm{Let}\mathrm{a}\mathrm{be}\mathrm{the}\mathrm{first}\mathrm{term}\mathrm{and}\mathrm{r}\mathrm{be}\mathrm{the}\mathrm{common}\mathrm{ratio}. \\ \mathrm{Hence}\mathrm{series}\mathrm{will}\mathrm{be}\mathrm{a},\mathrm{ar},{\mathrm{ar}}^2,{\mathrm{ar}}^3.... \mathrm{upto} \mathrm{infinity} \\ \mathrm{According} \mathrm{to} \mathrm{question}: \\ \mathrm{a}+\mathrm{ar}=15 \\ \Rightarrow \mathrm{a}\left(1+\mathrm{r}\right)=15 \\ \mathrm{a}=\frac{15}{1+\mathrm{r}} ;\left(\mathrm{i}\right) \\ \mathrm{Also} \\ \mathrm{a}=\mathrm{ar}+{\mathrm{ar}}^2+{\mathrm{ar}}^3....\mathrm{upto} \mathrm{infinity} \\ \mathrm{Dividing} \mathrm{both} \mathrm{sides} \mathrm{by} \mathrm{a} \mathrm{we} \mathrm{get}, \\ 1=\mathrm{r}+{\mathrm{r}}^2+{\mathrm{r}}^3+.....\mathrm{upto} \mathrm{infinity} \\ \mathrm{Sum} \mathrm{of} \mathrm{infinite} \mathrm{terms} \mathrm{of} \mathrm{G}.\mathrm{P}.=\frac{\mathrm{First} \mathrm{Term}}{\mathrm{Common} \mathrm{Ratio}} \\ \Rightarrow 1=\frac{\mathrm{r}}{1-\mathrm{r}} \\ \Rightarrow 1-\mathrm{r}=\mathrm{r} \\ \Rightarrow 2\mathrm{r}=1 \\ \Rightarrow \mathrm{r}=\frac{1}{2} \\ \mathrm{Hence} \mathrm{in} \left(\mathrm{i}\right), \mathrm{we} \mathrm{get}, \\ \mathrm{a}=\frac{15}{1+{\displaystyle \frac{1}{2}}}=10 \\ \mathrm{Hence} \mathrm{series} \mathrm{is} 10, 10\left(\frac{1}{2}\right), 10{\left(\frac{1}{2}\right)}^2, 10{\left(\frac{1}{2}\right)}^3,.....\mathrm{upto} \mathrm{infinity} \\ =\boxed{\mathbf{10}\mathbf{,}\mathbf{ }\mathbf{5}\mathbf{,}\mathbf{ }\frac{\mathbf{5}}{\mathbf{2}}\mathbf{,}\mathbf{ }\frac{\mathbf{5}}{\mathbf{4}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{upto}\mathbf{ }\mathbf{infinity}} \end{gathered}$$

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