The need to combine resistors in series or in parallel occurs so frequently that it warrants special attention. The process of combining the resistors is facilitated by combining two of them at a time. With this in mind, consider the single-loop circuit of Fig. 2.29. 

Figure 2.29 A single-loop circuit with two resistors in series.

The two resistors are in series, since the same current i flows in both of them. Applying Ohm’s law to each of the resistors, we obtain:


If we apply KVL to the loop (mo ving in the clockwise direction), we have 
Combining Eqs. (2.24) and (2.25), we get: 


Notice that Eq. (2.26) can be written as: 

implying that the two resistors can be replaced by an equivalent resistor Req; that is,


Thus, Fig. 2.29 can be replaced by the equivalent circuit in Fig. 2.30. The two circuits in Figs. 2.29 and 2.30 are equivalent because they exhibit the same voltage-current relationships at the terminals a-b. An equivalent circuit such as the one in Fig. 2.30 is useful in simplifying the analysis of a circuit. 
Figure 2.30 Equivalent circuit of the Fig. 2.29 circuit.
In general, 

The equivalent resistance of any number of resistors connected in series is the sum of the individual resistances

For N resistors in series then: 

To determine the voltage across each resistor in Fig. 2.29, we substitute Eq. (2.26) into Eq. (2.24) and obtain: 
Notice that the source voltage v is divided among the resistors in direct proportion to their resistances; the larger the resistance, the larger the voltage drop. This is called the principle of voltage division, and the circuit in Fig. 2.29 is called a voltage divider. In general, if a voltage divider has N resistors (R1, R2, … ,RN) in series with the source voltage v, the nth resistor (Rn) will have a voltage drop of: