A circuit has to be constructed with an unlimited number of 100 ohm resistors and a 10 V battery so that the circuit current is 500 mA. When a voltage V is applied across a resistor R the current flowing through it is I where V = I*R.
If the total current that flows has to be 500 mA, the equivalent resistance R = 10/0.5 = 20 ohm
If 2 resistors R1 and R2 are connected in parallel, the equivalent resistance R is given by the equation 1/R = 1/R1 +1/R2.
Using n 100 ohm resistors are used in parallel gives n/100= 1/20
=> n= 100/20 = 5
But here, the power through each resistor is 10*0.1 = 1 W which exceeds the capacity of the resistor.
To reduce the power to 0.25 W, the maximum current that can flow through each resistor when 10 V is applied across it is 0.025 A. This requires the use of 0.5/0.025 = 20 chains of resistors in parallel. The equivalent resistance of each of these is 10/0.025 = 400 ohm. To create a resistor with equivalent resistance 400 ohm, 4 100 ohm resistors should be connected in series.
The required circuit consists of 20 chains of 4 100 ohm resistors connected in series, each connected in parallel with 10 V applied across them.