How do we calculate the equivalent resistance of an infinite ladder if all the resistances involved are not equal? actually I know how do we calculate the equivalent resistance of an infinite ladder network if all the resistors have the same resistance. But this particular question has just puzzled me. Please help me with this and tell me the approach.
 A: This is more of a mathematical question than a physics one.
In principle you approach these kind of problems by writing down an equation that models the system in a recursive way. And then solve for the unknown(s).
You start by an appropriate definition. This is often the key. Lets say 
$S_n = \text{<resistance of the network when the "first" resistance is $2^n \Omega$}$
Clearly you search for $S_0$ as this is resistance between A and B in your drawing.
Now we can express $S_0$ with the help of $S_1$:
$S_0 = 1\Omega + 1 \Omega + (1 \Omega || S_1)$
Now, when you think about it, you will notice that $S_1$ and $S_0$ are related.
We have 
$S_1 = 2 S_0$
since all values resistance values for the network corresponding to $S_1$ are twice as large as those for $S_0$. This makes the total resistance ($S_1$) also twice as large.
This should give you enough information to be able to solve this on your own.
In case there is no easy connection, such as the one between $S_0$ and $S_1$ one would go on and try to establish a general recursion formula and try to solve it with mathematical methods. This, however, does not need to be easy. It is just the systematic way.
