Question

The first term of an infinite G.P. is $1$ and any term is equal to the sum of all the succeeding terms. Find the series.

Answer 1

Let the G.P. be $1,r,r^2,r^3,...$
given condition $\Rightarrow$ $r=\frac{r^2}{1-r}$ $\Rightarrow$ $r=\frac{1}{2}.$
Hence series is $1,\frac{1}{2},\frac{1}{4},\frac{1}{8},...,\infty$