The circuit diagram shown in the figure consist of a large number of elements (each element has two resistors `R_1 and R_2`). The resistance of the resistors in each subsequent element differs by a factor of `k= 1/2` from the resistance of the resistors in previous elements. Find the equivalent resistance between A and B.

Options

A. `R_1 - R_2`
B. `((R_1^2 - R_2) + sqrt(6R_1R_2))/(2)`
C. `((R_1 - R_2) + sqrt(R_1^2 +R_2^2 +6R_1R_2))/(2)`
D. none of these

Answer

C

Explanation

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) +(kXR_(2))/(kX+R)`
For `k = (1)/(2), X = ((R_(1) - R_(2)) +sqrt(R_(1)^(2) +R_(2)^(2) + 6R_(1)R_(2)))/(2)`