You are given a circuit made up of infinite numbers of repeated sections with resistance $R_{1}$ and $R_{2}$ as shown in fig. Find the effective resistance between the points A and B

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$4 \sqrt{R_{1} + R_{2}}$

$\frac{R_{1}}{2} + \frac{4 R_{2}}{R_{1}}$

$\frac{R_{1}}{2} + \sqrt{4 R_{1} R_{2}}$

$\frac{R_{1}}{2} [1 + \sqrt{1 + \frac{4 R_{2}}{R_{1}}}]$

answer is A.

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Detailed Solution

$R_{e q} = R_{1} + \frac{R_{2} R_{e q}}{R_{2} + R_{e q}}$
$\begin{array}{l} \Rightarrow R_{e q}^{2} - R_{1} R_{e q} - R_{1} R_{2} = 0 \\ ∴ R_{e q} = \frac{r_{1} + \sqrt{R_{1}^{2} + 4 R_{1} R_{2}}}{2} \\ = \frac{R_{1}}{2} [1 + \sqrt{1 + \frac{4 R_{2}}{R_{1}}}] \end{array}$