I am trying to find a closed form expression to allow me to solve for the nth term of the function $C(n)$, which I’ve written out below. Currently, the function for $C(n)$ is recursive, as one must solve for $C(n-1), C(n-2), C(n-3),{\cdot\cdot\cdot}$ in order to solve for $C(n)$. I have tried a bunch of different approaches, and am at the limits of my math capabilities (Bard and ChatGPT couldn’t help either). Please help me solve this! (or tell me if it’s not possible to express the following as a closed form expression.)
$$C(n) = C(0) - 2 \cdot V \cdot (1-F)\cdot\sum_{i=0}^{n-1}\frac{C(i) }{C(i) + Q \cdot V\cdot i}$$
At risk of over-explaining:
- $C(n)$ is the $n$th term of the function
- $C(0)$ is the initial condition of the function (this will be known)
- $F$ is a constant
- $V$ is a constant
- $Q$ is a constant
- $i$ is the $i$th term of the summation term, which ranges from $0$ to $n-1$