The equivalent resistance between points A and B of an infinite network of resistance each of 1Ω connected as shown is

The equivalent resistance between points A and B of an infinite network of resistance each of $1 Ω$ connected as shown is

A
infinite
B
$2 Ω$
C
zero
D
$(\frac{1 + \sqrt{5}}{2})$

Solution:

$R_{A B} = R + \frac{R \times R_{A B}}{R + R_{A B}}$

$R_{A B} = 1 + \frac{1 \times R_{A B}}{1 + R_{A B}}$

$R_{A B} (1 + R_{A B}) = 1 + 2 R_{A B}$

$R_{A B} + R_{A B}^{2} = 1 + 2 R_{A B}$

$R_{A B}^{2} - R_{A B} - 1 = 0$

$\frac{+ 1 \pm \sqrt{1 + 4}}{2} = \frac{1 + \sqrt{5}}{2}$

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