What does a duplicate-triplicate-etc ratio mean? So, if I have three numbers such that : $\dfrac ab = \dfrac bc$ Then we have $\dfrac ac$ which is a duplicate ratio of of $\dfrac ab$ If we have $4$ numbers such that : $\dfrac ab=\dfrac bc=\dfrac cd$ Then we have $\dfrac ad$ is triplicate ratio of $\dfrac ab$ We can continue this way with other proportions. So, what does it mean exactly ?

If you want to find such a $n$-tuple of numbers $a_1, a_2, \ldots, a_n$ such that $$\frac{a_1}{a_2} = \frac{a_i}{a_{i+1}} \qquad \forall\ 2\le i\le n-1$$ and also $a_i \in\mathbb N\quad\forall\ 1\le i\le n$, there is a simple algorithm:

Chose numbers $k,r\in\mathbb N$ and let

$$a_i = kr^{i-1}$$

The constant ratio will be $\frac1r$ in this case and all $a_i$ will be natural numbers.

As to what you can do with it: There is not much to it and I've never heard of the definitions you gave.

For an actual example, chose $k = 1, r = 2$ then we get the boring sequence $$\frac12 = \frac24 = \frac48 = \frac8{16} = \ldots$$