For parallel circuits, the potential drop across each resistor, R1, R2, ... is same i.e. V. Now if we want to replace them with Req, the potential drop across it would also be V. Using this information, we can write for the current in circuit as,
Ieq = V/Req, I1 = V/R1, I2 = V/R2 and so on.
But Ieq = I1 + I2 + I3
V/Req = V/R1 + V/R2 + V/R3
1/Req = 1/R1 + 1/R2 + 1/R3
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