what's the resistance of an infinite resistance ladder?

Question

Connect two 1 ohm resistors in series. Across the second connect another pair of 1 ohm reistors. Across the second o those two connect a third pair. Keep rpeating indefinitely. Now measure the reistance across both of the original resistors. What resistance will you measure?

Answer 1

If you look into the network connected across the second of the original two resistors, you will see the same infinite ladder that you're trying to mesure. If the answer is R, the entire ladder looks like a 1 ohm resistor in series with the parallel connection of R and 1 ohms. Therefore R = 1 + R/(1+R)' The solution is R=(1+sqrt(5))/2 or 1.618 ohms.

Try it for yourself.

1+ 1/(1+1/(1+1/(1+1/(...)))) --> 1.618