Map of increasing positive integer sequence to the rationals

Map of increasing positive integer sequence to the rationals

This is related to my answer of Limit of recursive sequence $a_{n+1} =
\frac{a_n}{1- \{a_n\}}$ and seems to be an interesting question.
Let $0 < a_1 < a_2 < \ldots < a_n$ be an increasing sequence of $n$ integers.
Define $c(a) = 1 + a_n[1 + (1+a_{n-1})[1 + (1 + a_{n-2})[1 + \ldots
(1+a_3) [ 1 + (1 + a_2)]]\ldots ]$
and $d(a) = a_n (1+a_{n-1})(1+a_{n-2})\cdots (1+a_2)(1+a_1)$.
The definition of $c$ seems a bit ambiguous for the beginning, so let's
specify $c(a_1) = 1$ and $c(a_1,a_2) = 1 + a_2$, and $c(a_1,a_2,a_3) = 1 +
a_3(1+(1+a_2))$. Also $d(a_1) = a_1$ and $d(a_2) = a_2(1+a_1)$.
Is the mapping $a \mapsto d(a)/c(a)$ injective as a mapping from $X_n \to
\mathbb{Q}$ where $X_n$ is a strictly increasing positive integer sequence
of length $n$? As $n$ tends to $\infty$ what is the range?
For example, with $a_i = i$, we have $\frac{n n!}{1+n(1 +
n(1+6(1+5(1+(4)(1+3)))\ldots)}$
$ \frac{1}{1}, \frac{2*2}{1+2}=\frac{4}{3},
\frac{3*3*2}{1+3(1+3)}=\frac{18}{13},
\frac{4*4*3*2}{1+4(1+4(1+3)))}=\frac{96}{69},\ldots $
I guess it would also be nice to prove whether the remainder sequence of
the other problem on these rationals gives $a_n$ as the limit, but that
might be more appropriate for that thread.