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A network of resistance is constructed with $R 1 and R 2$ as shown in figure. The potential at the points 1,2,3,…….,N are $V 1, V 2, V 3, … V n$ respectively each having a potential k times smaller than the previous one . Find ratio $\frac{R_{1}}{R_{2}}$ .
(Example $V 0 = k V 1,, V 1 = k V 2, V 2 = k V 3$

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k 2 − 1 k

k k − 1

k − 1 k 2

(k − 1) 2 k

answer is D.

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Detailed Solution

$\frac{V_{0} - V_{1}}{R_{1}} = \frac{V_{1} - V_{2}}{R_{1}} + \frac{V_{1} - 0}{R_{2}}$

$Substituting V_{1} = \frac{V_{0}}{k}, V_{2} = \frac{V_{1}}{k} = \frac{V_{0}}{k^{2}}$

$\frac{V_{2} k^{2} - V_{2} k}{R_{1}} = \frac{V_{2} k - V_{2}}{R_{1}} + \frac{V_{2} k}{R_{2}}$

$\Rightarrow \frac{k^{2} - k}{R_{1}} = \frac{k - 1}{R_{1}} + \frac{k}{R_{2}}$

$\Rightarrow \frac{R_{1}}{R_{2}} = \frac{(k - 1)^{2}}{k}$

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