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(C) Let the effective resistance between $A$ and $B$ be $R$. Since the network is infinite,adding or removing one repeating unit will not change the t

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$(1 + \sqrt{3}) \, \Omega$

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(C) Let the effective resistance between $A$ and $B$ be $R$. Since the network is infinite,adding or removing one repeating unit will not change the total resistance. The circuit can be viewed as a $1 \, \Omega$ resistor in series with the top wire,a $1 \, \Omega$ resistor in series with the bottom wire,and a $1 \, \Omega$ shunt resistor,followed by the rest of the infinite network which also has an equivalent resistance of $R$. Thus,the equivalent resistance $R$ is given by: $R = 1 + 1 + \frac{1 \cdot R}{1 + R}$ $R = 2 + \frac{R}{1 + R}$ $R(1 + R) = 2(1 + R) + R$ $R + R^2 = 2 + 2R + R$ $R^2 - 2R - 2 = 0$ Using the quadratic formula $R = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$: $R = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-2)}}{2(1)}$ $R = \frac{2 \pm \sqrt{4 + 8}}{2} = \frac{2 \pm \sqrt{12}}{2} = \frac{2 \pm 2\sqrt{3}}{2} = 1 \pm \sqrt{3}$ Since resistance must be positive,we take $R = 1 + \sqrt{3} \, \Omega$.

What is the effective resistance between $A$ and $B$ for the given infinite resistor network?

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