GATE EC 2014 Set 4 — Question 42

The problem asks for the equivalent resistance of a non-uniform infinite ladder network.
First, let's find the equivalent resistance, let's call it $R_{e'}$, of a uniform infinite ladder where each series element is R and each shunt element is R.
Due to the infinite nature, the resistance looking into the ladder is the same even after adding one section. So, $R_{e'}$ can be written as:
$R_{e'} = R + (R \|\| R_{e'}) = R + \frac{R \cdot R_{e'}}{R + R_{e'}}$
Multiplying by $(R + R_{e'})$:
$R_{e'}(R + R_{e'}) = R(R + R_{e'}) + R \cdot R_{e'}$
$R \cdot R_{e'} + R_{e'}^2 = R^2 + R \cdot R_{e'} + R \cdot R_{e'}$
$R_{e'}^2 - R \cdot R_{e'} - R^2 = 0$
This is a quadratic equation for $R_{e'}$. Using the quadratic formula and taking the positive root for resistance:
$R_{e'} = \frac{-(-R) + \sqrt{(-R)^2 - 4(1)(-R^2)}}{2} = \frac{R + \sqrt{R^2 + 4R^2}}{2} = \frac{R + \sqrt{5R^2}}{2} = R \frac{1+\sqrt{5}}{2}$Now, let's analyze the given network. The first series resistor is 2R, and the rest of the network is a shunt resistor R in parallel with a uniform infinite ladder (whose resistance is $R_{e'}$).
The total equivalent resistance $R_e$ is:
$R_e = 2R + (R \|\| R_{e'})$
$R_e = 2R + \frac{R \cdot R_{e'}}{R + R_{e'}}$
From our earlier equation for $R_{e'}$, we know that $\frac{R \cdot R_{e'}}{R + R_{e'}} = R_{e'} - R$.
Substituting this into the expression for $R_e$:
$R_e = 2R + (R_{e'} - R) = R + R_{e'}$
Now, substitute the value of $R_{e'}$:
$R_e = R + R \frac{1+\sqrt{5}}{2} = R \left(1 + \frac{1+\sqrt{5}}{2}\right) = R \left(\frac{2+1+\sqrt{5}}{2}\right) = R \frac{3+\sqrt{5}}{2}$The question asks for the value of $R_e/R$.
$\frac{R_e}{R} = \frac{3+\sqrt{5}}{2}$
Using the value $\sqrt{5} \approx 2.236$:
$\frac{R_e}{R} \approx \frac{3+2.236}{2} = \frac{5.236}{2} = 2.618$
This value is within the specified range of 2.60 to 2.64.