Question

what is the relationship between the currents through the individual resistance and current through the entire series network?

Answer 1

When the resistors are connected in series  

The potential difference across the individual resistors are dV1 and dV2 then the net potential difference acrpss the combination is  

dV = dV1+dV2

by definition of potential is work done per unit charge

the net work done in the first resistor and work done in the second and third resistor and  

the currents in the both resistors are exactly the same, since anby charge that flows through the first resistor continues to flow through the second.

so From Ohm's law we can write

dV = dV1+dV2 = I*R1+I*R2 = I(R1+R2)

so the current through the individual resistance and the cuirrent through the entire series network will be same.