As shown in the figure, a finite ladder is assembled by connecting...

For the equivalent capacitance of the ladder between points A and B to be independent of the number of sections, the equivalent capacitance of the entire ladder must be equal to the terminating capacitance $C$.

Let the equivalent capacitance of the ladder be $C_{eq}$. By observing the repeating unit of the circuit from the terminals A and B, the ladder consists of a $2\mu\text{F}$ capacitor in parallel with the series combination of a $4\mu\text{F}$ capacitor and the rest of the ladder.

Thus, the equivalent capacitance is given by:

$C_{eq} = 2 + \dfrac{4 \times C_{eq}}{4 + C_{eq}}$

Since the equivalent capacitance is independent of the number of sections, we must have $C_{eq} = C$. Substituting this into the equation:

$C = 2 + \dfrac{4C}{4 + C}$

Multiplying both sides by $(4 + C)$:

$C(4 + C) = 2(4 + C) + 4C$

$4C + C^2 = 8 + 2C + 4C$

$C^2 - 2C - 8 = 0$

Factorizing the quadratic equation:

$(C - 4)(C + 2) = 0$

Since capacitance cannot be negative, we get $C = 4\mu\text{F}$.