How do you combine resistors is series and parallel?

In a series circuit each resistor will have the same current passing through it.  If we assume that the wires connecting the resistors have no resistance, then the total potential difference, V, is given by Kirchhoff’s Voltage Law;

V=V₁+V₂+V₃+…

Using Ohm’s Law,

V=IR,

we get

IR=IR₁+IR₂+IR₃+…

as the current is the same in each resistor we can cancel them out to get;

R=R₁+R₂+R₃+…

Basically if you have 2 or more resistors in series the total resistance is simply the sum of each resistance.

Combining resistances in parallel is a bit more complicated; in a parallel circuit each resistor will have the same potential difference, but the current flowing through them will not necessarily be the same.  If we start with the conservation of charge we get;

I=I₁+I₂+I₃+…

using Ohm’s Law again gives us

V/R=V/R₁+V/R₂+V/R₃+…

and as V is the same for all the resistors we get

1/R=1/R₁+1/R₂+1/R₃+…

If you have 2 or more resistors in parallel the reciprocal of the resistance (1 over the resistance) is the sum of the reciprocals of each resistance.