Context:
My problem is analogous to having $n \in \mathbb{N}^*$ priority levels to which I want to associate exponential levels of ressources. The total ressources allocated must be 100% (ideally, this would be exact). Also, I'd like to fix the portion of ressources allocated to the first level (in order to be sure that it wouldn't be set to $0$). As I want the sum of portions to equal 1, this needs to be comprised between $0$ and $\frac 1 n$. If possible, I would also like to be able to tune the growth of the exponential once the first point has been fixed.
What I've done
I've defined my function $f(x) = c_1 \cdot e^{c_2x} + c_3$ and have summed up my constraints as $(1): c_1 + c_3 = y_0$ and $(2): \sum_{i=0}^{n-1} f(i) = 1$.
I also formalized my problem as follow:
Find $T(y_0) \subset \mathbb{R}^3$ such that: $\forall y_0 \in [0, \frac 1 n]$, $\forall (c_1, c_2, c_3) \in T(y_0)$,
$
\left\{\begin{array}{@{}l@{}}
c_1 + c_3 = y_0\\
\sum_{i=0}^{n-1} c_1 \cdot e^{c_2x} + c_3 = 1
\end{array}\right.\,
$
What I need
I hope the summary of the problem into the constraints is correct, if not, please do not hesitate to point it out.
Also, if for any reason this would not be possible, this would also be something interesting!
If the first two points are OK, the thing I would need most is help to solve the problem!
Thanks to everyone who will take the time to read!
EDIT
Thanks to @abiessu's comments and by removing $c_3$ (not as useful as I expected it to be), I've obtained $(2^3)$: $\frac {1 - e^{9 \cdot c_2}} {1 - e^{c_2}} = \frac {1} {c_1}$, but I am stuck here at the moment, and don't know how to get $c_2$ as a function of $c_1$.
EDIT 2
By substituting $X=e^{c_2}, c_2 \neq 0$, we get $(2^4)$: $\frac {1 - X^9} {1 - X} = \frac {1} {c_1}$. Let's define $g(x)=\frac {1 - x^9} {1 - x}$.
There should be some solution to this problem, seeing as $g$ is defined on $[\sqrt[9]1; 1[$ (not the whole domain), and that $g(\sqrt[9]1)=0$ and $lim_{x \rightarrow 1}g(x) = +\infty$. So, according to the Intermediate Value Theorem, $\forall c_1 \in ]0; +\infty[, \exists c_2 \in [e^{\sqrt[9]1}; e^1]$ s.t. $\frac {1 - e^{9 \cdot c_2}} {1 - e^{c_2}} = \frac {1} {c_1}$.
This is nice to know (if it is even correct), but this isn't making me closer to the solution...