Before going through these derivations it is advertised that you read up on Kirchhoff’s two laws and read through resistors in series and parallel first.

Using the following circuit setup and Kirchhoff’s laws, the equation for resistors in series can be derived;

Kirchhoff’s voltage law (KCL) tells us that   \epsilon = V_{1} + V_{2}  (1)

Since  V = IR we can write;

V_{1} \rightarrow I_{1}R_{1}
V_{2} \rightarrow I_{2}R_{2}
\epsilon \rightarrow I_{3}R_{T}
The power source is providing power to the entire circuit which consists of ALL the resistors, R_{T}

So we can now sub these into equation (1) and write;

I_{3}R_{T} = I_{1}R_{1} + I_{2}R_{2}  (2)

Kirchhoff’s current law (KCL) tells us that because this is a series circuit the current is the same through both resistors and so;

I_{1} = I_{2} = I_{3} = I

Therefore equation (2) becomes;

IR_{T} = IR_{1} + IR_{2}

Divide the whole equation by I to give:

R_{T} = R_{1} + R_{2}

Expanding this equation for more than just two resistors in series and we can write the full equation;

R_{T} = R_{1} + R_{2} + R_{3} + ...


Further reading: