I was trying out a physics question to find the equivalent resistance between two points A and B.
I got a functional equation on solving, but I couldn't proceed further. Finding f(1) gives the answer to the question.
\begin{gather*} \frac{( x+1) f( x+2)}{( x+1) +f( x+2)} \ +\ x=f( x) \ \\ \end{gather*}
Is it possible to solve it, or should I consider a different approach?
First consider a finit ladder (limited to $n$ rungs) drawing below.
If n is sufficiently large the equivalent resistance $R_0$ is quite undependent from the last resistor i.e. what ever $R_n$ be from $0$ to infinity. This is easy to check with numerical recursive calculus thanks to the formula below.
For an infinite ladder the limit found is : $$\boxed{R_0\simeq 2.485339738238447...}$$
Of course the value found is not analytic but numerical. So my answer isn't a final answer.
Hint : Thanks to ISC (Inverse Symbolic Calculator) one can guess a probable analytic formula : $$R_0=\frac{4}{\ln(5)}\simeq 2.48533973823844724...$$ The councidence is correct for more than 16 digits. The proposed formula can be considered as a good conjecture.
The above result was obtained with my own ISC : https://fr.scribd.com/doc/14161596/Mathematiques-experimentales
Similar studies about ladders made of not only resistors but mixed resistors and capacitors : https://fr.scribd.com/doc/71923015/The-Phasance-Concept