Question

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

Answer 1

Given that $T_p=(T_{p+1}+T_{p+2}+...\infty )ora{r}^{R-1}=a{r}^p+a{r}^{p+1}...\infty .$
But $a=1\therefore r^{p-1}=\frac{r^p}{1-r}$
$\therefore 1-r=r$ or $r=\frac{1}{2}$
Hence the series is $1,\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16}...\infty$
And sum $=\frac{1}{1-\frac{1}{2}}=2$