<s>[INST] Is there a closed form for the recurrence $V_{n+1}={V_n+\Delta V\over 1+{V_n\cdot \Delta V/C^2}}$, for constants $\Delta V$ and $C$? I was wondering if the following recurrence formula has a closed form:

$$V_{n+1}={V_n+\Delta V\over 1+{V_n\cdot \Delta V\over C^2}}$$

where $\Delta V$ and $C$ are positive constants, $V_n$ is the velocity of the $n$-the inertial frame and the primary velocity $V_0$ is given (take it $0$ if needed).
Attempt
This sequence obviously tends to $C$ (the speed-of-light supremum of speeds of observations), so I naturally tried to crack it using $$e_n=V_n-C$$but I failed. Any idea is appreciated.
Note
The above rule determines the Relativistic Velocity-addition Formula where $V_n$ is supposed to be the velocity of the inertial frame $2$ that is moving with respect to us (inertial frame $1$) and $\Delta V$ is an increase in the speed of the moving object (or we can assume it as the relative speed of object in the inertial frame 2). My work basis is the Lorentz Transformation.
 [/INST] As I have noted in the comment, Wolfram Alpha gives the closed form, with $k = \Delta V$:
$$V_n = \frac{C_1 c(c+k) \left( \frac{1}{k-c} + \frac{1}{c}\right)^n - c(c-k)\left(-\frac{k}{c(c+k)}\right)^n}{(c-k)\left(-\frac{k}{c(c+k)}\right)^n + C_1 (c+k) \left(\frac{1}{k-c}+\frac{1}{c}\right)^n}$$
Taking $V_0 = 0$, we can see
$$ 0 = \frac{C_1 c(c+k)  - c(c-k)}{(c-k) + C_1 (c+k) }$$
which is undefined if $C_1 = -\frac{c-k}{c+k}$, so let $C_1 \ne -\frac{c-k}{c+k}$, then
$$0 = C_1 c(c+k)-c(c-k)\Rightarrow 0 = C_1(c+k)-c+k \Rightarrow C_1=\frac{c-k}{c+k}$$
Hence $C_1$ is undefined when $k = \Delta V = -C$, so depending on the physical context of special realtivity (which I don't know enough of), this may never happen
Substituting this into the original closed form:
\begin{align}V_n &= \frac{(\frac{c-k}{c+k}) c(c+k) \left( \frac{1}{k-c} + \frac{1}{c}\right)^n - c(c-k)\left(-\frac{k}{c(c+k)}\right)^n}{(c-k)\left(-\frac{k}{c(c+k)}\right)^n + (\frac{c-k}{c+k}) (c+k) \left(\frac{1}{k-c}+\frac{1}{c}\right)^n} \\
&=\frac{c(c-k)\left[ \left( \frac{1}{k-c} + \frac{1}{c}\right)^n - \left(-\frac{k}{c(c+k)}\right)^n\right]}{(c-k)\left[\left(-\frac{k}{c(c+k)}\right)^n +   \left(\frac{1}{k-c}+\frac{1}{c}\right)^n\right]}\\
&= c \frac{\left[ \left( \frac{1}{k-c} + \frac{1}{c}\right)^n - \left(-\frac{k}{c(c+k)}\right)^n\right]}{\left[\left(-\frac{k}{c(c+k)}\right)^n +   \left(\frac{1}{k-c}+\frac{1}{c}\right)^n\right]}\end{align}
Hence in terms of the way you formulated you have
$$ \boxed{V_n=C \frac{\left[ \left( \frac{1}{\Delta V-C} + \frac{1}{C}\right)^n - \left(-\frac{\Delta V}{C(C+\Delta V)}\right)^n\right]}{\left[\left(-\frac{\Delta V}{C(C+\Delta V)}\right)^n +   \left(\frac{1}{\Delta V-C}+\frac{1}{C}\right)^n\right]} \quad \Delta V \ne \pm C ,\, C \ne 0}$$
if one wishes a preferable altenartive form without negatives inside the power terms:
\begin{align}V_n&=C \frac{\left[ \left( \frac{\Delta V}{C(\Delta V - C)}\right)^n - \left(-\frac{\Delta V}{C(C+\Delta V)}\right)^n\right]}{\left[\left(-\frac{\Delta V}{C(C+\Delta V)}\right)^n +   \left(\frac{\Delta V}{C(\Delta V - C)}\right)^n\right]} \\
&=C \frac{\left[ \left( -\frac{\Delta V}{C(C-\Delta V)}\right)^n - \left(-\frac{\Delta V}{C(C+\Delta V)}\right)^n\right]}{\left[\left(-\frac{\Delta V}{C(C+\Delta V)}\right)^n +   \left(-\frac{\Delta V}{C(C- \Delta V)}\right)^n\right]} \\
&=C \frac{\left[ (-1)^n \left(\frac{\Delta V}{C}\right)^n\left( \frac{1}{(C-\Delta V)}\right)^n - (-1)^n \left(\frac{\Delta V}{C}\right)^n \left(\frac{1}{(C+\Delta V)}\right)^n\right]}{\left[(-1)^n \left(\frac{\Delta V}{C}\right)^n\left(\frac{1}{(C+\Delta V)}\right)^n +   (-1)^n \left(\frac{\Delta V}{C}\right)^n \left(\frac{1}{(C- \Delta V)}\right)^n\right]} \end{align}
which is written
\begin{align}V_n&=C \frac{\left[ \left( \frac{1}{C-\Delta V}\right)^n - \left(\frac{1}{C+\Delta V}\right)^n\right]}{\left[\left( \frac{1}{C-\Delta V}\right)^n + \left(\frac{1}{C+\Delta V}\right)^n\right]} \\
&=C \frac{\left[ \left( \frac{1}{C-\Delta V}\right)^n - \left(\frac{1}{C+\Delta V}\right)^n\right]}{\left[\left( \frac{1}{C-\Delta V}\right)^n + \left(\frac{1}{C+\Delta V}\right)^n\right]} \frac{(C+\Delta V)^n (C-\Delta V)^n}{(C+\Delta V)^n (C-\Delta V)^n}\end{align} 
Finally,
$$\boxed{V_n=C \frac{\left[ \left( C+\Delta V\right)^n - \left(C- \Delta V\right)^n\right]}{\left[\left( C+ \Delta V\right)^n + \left(C- \Delta V\right)^n\right]} \quad \Delta V \ne \pm C ,\, C \ne 0}$$
which is in the form:
$$V_n = C \frac{a^n-b^n}{a^n+b^n}$$
which has limit $C$ as $n \to \infty $ (see this if you don't know how to show that), matching the limit in the OP's question.
If anyone knows how to derive the solution Wolfram gets, that would be preferable to this
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