Given an induction definition, how to calculate elements?

I'm having difficulty with a mathematical problem.

I've got the following;

The basis is:

 -1 ∈ V

And the induction is;

x ∈ V → x/(1-x) ∈ V

Now, I have made 4 statements and I want to get to know if they're either true, or false.

The statements are;

1. All elements are negative. (under zero)
2. All elements are between -2 and 0
3. -1/7 ∈ V 
4. -2/3 ∈ V 

There's also an exception;
There are no elements of V that cannot be acquired by applying statement 1 and 2.

Now, I'm not really sure how to start here. How can I effectively sort out elements in a induction definition?

Answer

You can proceed as follows :

1. Start by computing the first few terms : $x_0 = -1$, then
   $$x_1 = -1/(1+1) = -1/2$$
   $$x_2 = -1/2/(1+1/2) = -1/3$$
   $$x_3 = -1/3/(1+1/3) = -1/4$$
   
   This gives you an idea as to what $x_n$ should be. ie. $x_n = -1/(n+1)$




2. Prove your guess by induction :

a) $x_0 = -1 = -1/(0+1)$. Hence your guess is true when $n=0$

b) Assume $x_{n-1} = -1/n$, then
$$x_n = -1/n/(1+1/n) = -1/(n+1)$$

Hence, your guess is true for all $n\in \mathbb{N}$.

By your exception, it follows that
$$V = \{-1, -1/2, -1/3, \ldots \}$$