Question

Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.

Answer 1

a = 1

$a_n=a_{n+1}+a_{a+2}+a_{n+3}+....$

$a{r}^{n-1}+a{r}^n+a{r}^{n+1}+a{r}^{n+2}+....$

$a{r}^{n-1}=a{r}^n(1+r+r^2+....\infty$

$1=r\left(\frac{1}{1-r}\right)$

$1-r=r$

$1=2r$

$r=\frac{1}{2}$

G.P. is $1,\frac{1}{2},\frac{1}{4},\frac{1}{8},.....\infty$

Thus the infinite.