Let us consider two sequences $a_{n} = \frac{c^{2n}}{2n}$ and $b_{n} = c^{2n}$, $n \in \Bbb{N}$ then $\frac{a_{n+1}}{a_{n}} \rightarrow c^{2}$ as $n \rightarrow \infty$ and $\frac{b_{n+1}}{b_{n}} \rightarrow c^{2}$ as $n \rightarrow \infty$.
So if we plot $n$ vs $\frac{a_{n+1}}{a_{n}}$, then it should converge to the value $c^2$ (similar with $b_{n}$).
We see that both sequences converge to the same value, so how to distinguish between those two sequences if we don't know a prior about $a_{n}, b_{n}$? $c$ is some positive quantity less than $1$
any idea?
If we reframe it: given two sets of data points $P$ and $Q$, how to tell which data follows which sequence? any graphical method?
You have candidate functional forms, so fit them using a regression. For example, suppose your data set is $\{2.03, 2.06045, 2.78848, 4.24545, 6.89462\}$. The best-fit exponential is $$b_n=0.972149e^{0.385244 x}$$ which has the wrong constant prefactor and still has a residual of $\approx0.2$ at $n=5$. But it matches $$a_n=\frac{2.03^n}{n}$$ perfectly, because that's how I concocted the numbers in the first place.