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

Question:

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

Solution:

Here, first term, = 1

Common ratio = r

$\therefore a_{n}=\left[a_{n+1}+a_{n+2}+a_{n+3}+\ldots \ldots \infty\right] \forall n \in N$

$\Rightarrow a r^{n-1}=a r^{n}+a r^{n-1}+\ldots \ldots \infty$

$\Rightarrow r^{n-1}=\frac{r^{n}}{1-r} \quad$ [Putting $a=1$ ]

$\Rightarrow r^{n-1}(1-r)=r^{n}$

$\Rightarrow 1-r=r$

$\Rightarrow 2 r=1$

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

Thus, the infinte G.P is $1, \frac{1}{2}, \frac{1}{4}, \ldots \infty$.