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

Given that $T_p=(T_{p+1}+T_{p+2}+...\infty )$
or ${ar}^{p-1}={ar}^p+{ar}^{p+1}+...\infty$ But $a=1$
$\therefore r^{p-1}=\frac{r^p}{1-r}$ Sum of an infinite $G.P.$
$\therefore 1-r=r$ or $r=\frac{1}{2}$
Hence the series is $\therefore 1,\frac{1}{2},\frac{1}{4},\frac{1}{16}+...\infty$