How to formulate a GP according to this question?

Question

Q. Each term of a geometric progression is $\frac{1}{x} th$ of the sum of all the terms following it. Find the commmon ratio of the progression in terms of $x$.

A. $\frac{x-1}{x}$

B. $\frac{x}{x+1}$

C. $\frac{x+1}{x+2}$

D. None

The problem is how to formulate a GP according to the question in the first place? If it were that each term was $\frac{1}{x} th$ of the previous term it would be super simple. But it says the sum of terms 'following it'. So does it mean an infinite GP? What am I missing here?

P.S. : I'm new to here so this may be already solved before, but I tried my best to search it everywhere. Apologies in advance.

Answer 1

Take any arbitrary term $x_i$. By hypothesis, $x_i=\displaystyle\frac{1}{x}\sum_{t=i+1}^n x_t$.

Again, $x_{i+1}=\displaystyle\frac{1}{x}\sum_{t=i+2}^n x_t$.

Thereby, $x_{i}-x_{i+1}= \frac{1}{x}x_{i+1}\implies \frac{x_{i+1}}{x_i}=\frac{x}{1+x}$ for any $i \in \mathbb{N}$