The circuit diagram shown in fig consists of a very large (infinite) number of elements . The resistances of the resistors in each subsequent element differ by a factor k from the resistance of the resistors in the previous element. Determine the resistance `R_(AB)` between points A and B if the resistances of the first element are `R_1 and R_2`

Explanation

From symmetry considerations , we can remove the first element from the circuit, the resistance of the remaining circuit between point C and D will be `R_(CD)= kR_(AB)` . Therefore , the equivalent circuit of the infinite chain will have the form shown in figure.

Thus
`R_(AB) = R_1 = (R_2(kR_(AB)))/(R_2 + (kR_(AB)))`
`R_(AB_[R_2 + KR_(AB))] = R_1 [R_2 + kR_(AB)] + kR_2 R_(AB)`
`R_2 R_(AB) + kR_(AB)^2 = R_1 R_2 + kR_1 R_(AB) + kR_2 R_(AB)`
or `kR_(AB)^2 + (R_2 + kR_1 - kR_2)R_(AB) - R_1 R_2 = 0`
`therefore R_(AB) = (-(R_2 - kR_1 - kR_2)pm sqrt((R_2 - kR_1 - kR_2)^2 + 4kR_R_2))/(2k)`
As resistance cannot be negative
`R_(AB) = (kR_1 + kR_2 - R_2+ sqrt((R_2 - kR_1 - kR_2)^2 + 4kR_1 R_2))/(2k)`