Adding the resistances
Three lines of Ohm's law and one fact about the circuit give the series rule, and it is the simplest rule in the chapter.
Take the current flowing through the circuit as I. Because it is a series arrangement, that same I flows through all three. Applying V = IR to each resistor in turn:
potential difference across R1 = I R1
potential difference across R2 = I R2
potential difference across R3 = I R3
When resistors are connected in series, the sum of the potential differences across them must equal the supply potential difference. So, writing the supply p.d. as V and the whole circuit's resistance as R:
V = I R = I R1 + I R2 + I R3
Resistors in series
R = R1 + R2 + R3
Divide every term by I and the series rule is what is left.
What "equivalent resistance" means
The equivalent resistance is the value of the single resistor that could be put in place of all the connected resistors without the rest of the circuit noticing. When resistors are connected in series, the equivalent resistance is equal to the sum of all their values.
Series always gives you more
Every resistor you add in series makes the total larger, because you are lengthening the single path the current has to take. The equivalent resistance of a series set is always bigger than the biggest resistor in it.