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An infinite resistor ladder is built from identical repeating…
Question
An infinite resistor ladder is built from identical repeating sections. In each section a series resistor of 8 ohm is followed by a shunt resistor of 6 ohm that connects the two rails. The ladder is infinite to the right. Find the equivalent resistance, in ohm, looking into the input terminals of the ladder (input across the first series resistor and the leftmost shunt).
✓ Verified answer: 12checked by our engine — not a guess
Step-by-step solution
Let X be the input resistance.
Because the ladder is infinite, removing the first section leaves an identical ladder of resistance X.
Thus X = R1 + (R2 parallel X) = R1 + R2*X/(R2+X).
Multiply out: X(R2+X) = R1(R2+X) + R2*X, i.e.
X^2 - R1*X - R1*R2 = 0.
So X = [R1 + sqrt(R1^2 + 4 R1 R2)]/2 (positive root).
With R1=8, R2=6: discriminant = 64 + 192 = 256, sqrt = 16, X = (8+16)/2 = 12 ohm.
Final answer12
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