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The circuit diagram shown in figure consists of a large number of elements (each element has two resistors). The resistance of the resistors in each subsequent element differs by a factor of K = 1/2 from the resistances of the resistors in the previous elements. The equivalent resistance between A and B shown in figure is

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$\frac{(R_{1} - R_{2}) + \sqrt{6 R_{1} R_{2}}}{2}$

none of these

$\frac{(R_{1} - R_{2}) + \sqrt{R_{1}^{2} + R_{2}^{2} + 6 R_{1} R_{2}}}{2}$

$\frac{R_{1} - R_{2}}{2}$

answer is C.

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Detailed Solution

When each element of the circuit is multiplied by a factor k, then the equivalent resistance also becomes K times. Let the equivalent resistance between A and B be x.

So the equivalent circuit becomes

$X = R_{1} + \frac{{kXR}_{2}}{kX + R_{2}} For k = \frac{1}{2}, X = \frac{(R_{1} - R_{2}) + \sqrt{R_{1}^{2} + R_{2}^{2} + 6 R_{1} R_{2}}}{2}$

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