In a series circuit, the equivalent resistance is the sum of the individual resistances. The voltage across the first resistor is divided by the current, which is the reciprocal of the sum of the reciprocals of the resistances. The total voltage is divided by the number of resistors.
In a parallel resistance circuit, the voltage drop is the same across all resistances. The current is always the same through each resistance. The equivalent resistance is equal to the sum of the resistances. The current does not depend on the resistances.