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Prove. The equivalent resistance R of n resistors R₁, R₂, ... , R_n connected in parallel is given by

Proof. We will prove it for three resistors. See Fig. 1.

The potential difference across the terminals of each resistor must be the same and be equal to V_ab. Denote the currents passing through the three resistors R₁, R₂, and R₃ by i₁, i₂ , and i₃. Then, by Ohm’s law

Now the sum of the currents i₁, i₂ , and i₃ going through the resistances must be equal to the line current :

3)      i = i₁ + i₂ + i₃

Substituting 2) into 3) gives                                                                                                    

or

Since

equation 5) becomes