Finding Resistance in Infinite Chain of Resistors

[GingerBread27]

Consider the above infinite chain of resistors. Calculate the effective resistance, R in ohm of the network between the terminals A and B given that each of the resistances labelled r=4180 ohm.

I've split the resistor and I've done R^2-Rr-r^2=0, solving for R and I don't get the right answer.

I've also cut the resistor and worked it out so I get an equation of R^2+2Rr-2r^2=0 and I get the wrong answer.

Since it is an infinite chain I thought it was possible to split the resistor and keep maybe the first two or three resistors.

Please Help!

 

[Davorak]

I did not know how to solve this one. It felt like I should have been able to though. So I looked it up this problem type to see how they were genraly sovled. These types of problems are called ladder circuits. This is a infinite ladder ciruit, below is a link that expalins how to sovle them in genral. I do not think ladder circuits are focused on much now adays in electrical engineering since op amps are easier to use and cheap. What class is this problem for if you don't mind me asking?
http://www.crbond.com/papers/ent2-3.pdf
The general apporach to solving these circuit seems to be to add another rung of reistors on the front end of the circuit. Since the chain of rungs are in Infinite this will not change the reistance, but will give you an equation for the reistance.

 

[Curious3141]

What do you mean by "splitting the resistor" ? At any rate, your first equation R^2 - Rr - r^2 = 0 is correct.

Basically, I would approach it by observing that :

r + r||R = R

where || refers to "parallel to".

That gives the first quadratic you got (the second one is wrong). The solution of the correct quadratic is related to the golden ratio.

What answer was the book expecting ? If it's not R = r(\frac{1 + \sqrt{5}}{2}), the book is wrong.

 

[GingerBread27]

This is for a homework assignment for Physics 2 with Calculus.