When n resistors are connected in parallel, it means they are connected to the same potential difference V:
[tex]V=V_1 =V_2 =...=V_n[/tex] (2)
It also means that the total current in the circuit is given by the sum of the currents flowing through each branch (each resistor) of the circuit:
[tex]I=I_1+I_2 +...+I_n[/tex] (1)
By using Ohm's law:
[tex]I= \frac{V}{R} [/tex]
we can rewrite (1) as
[tex] \frac{V}{R_{eq}} = \frac{V_1}{R_1}+ \frac{V_2}{R_2}+...+ \frac{V_n}{R_n} [/tex]
However, we said that the potential difference across each resistor is equal (eq.(2)), so we can rewrite the last formula as
[tex] \frac{V}{R_{eq}} = \frac{V}{R_1}+ \frac{V}{R_2}+...+ \frac{V}{R_n}[/tex]
From which we find an expression for the equivalent resistance of n resistors in parallel:
[tex] \frac{1}{R_{eq}}= \frac{1}{R_1}+ \frac{1}{R_2}+....+ \frac{1}{R_n} [/tex]
