1. The General Formula

For an infinite ladder with series resistance R_s and shunt resistance R_p, the equivalent resistance R is:

R = R_s + (R_p || R)

This leads to the quadratic equation: R² – R_sR – R_sR_p = 0.

2. Why the Answer is (c) (√5 + 1)r

The diagram shows “r” on the top and “r” on the bottom, making the total series resistance per section R_s = 2r.

The vertical shunt resistance is R_p = 2r (due to symmetry).

Substituting R_s = 2r and R_p = 2r:

R² – 2rR – (2r)(2r) = 0

R² – 2rR – 4r² = 0

Solving for R using the quadratic formula:

R = [2r ± √(4r² – 4(1)(-4r²))] / 2

R = [2r ± √(20r²)] / 2

R = r + r√5 = (√5 + 1)r