Rational functions serie

Question

I've stumbled upon an interesting question about rational functions I can't answer. What follows is given: $$f(x) = \frac{x}{1-x} $$ $$ x_1 = \frac{1}{a}, x_2 = f(x_1), x_3 = f(x_2), x_4 = f(x_3), ..., x_{23} = f(x_{22}), x_{24} = f(x_{23}) = 1$$ What is $a$ equal to? How can you compute this? (There is a solution!) Some context: it comes from a math Olympiad. I tried just filling it in, but there should be an easier way!

Answer 1

If I was lazy, I should compute the first terms which give the sequence $$\left\{\frac{1}{a},\frac{1}{a-1},\frac{1}{a-2},\frac{1}{a-3},\frac{1}{a-4},\frac{1 }{a-5},\cdots\right\}$$ and the pattern is clear.

If I was not lazy, I should solve the recurrence relation $$x_{n+1}=\frac {x_n} {1-x_n} \qquad \qquad x_1=\frac 1a$$ the solution of which is simple (I let you finding it).

So $a=24$