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Consider a circuit where 2 resistors are connected in parallel, which are in turn connected to another resistor in series. Let the resistance of the two resistors in parallel be [tex]R_1[/tex] and [tex]R_2[/tex]. The resistance of the resistor in series is [tex]R_3[/tex]

Normally we would add the resistance like this to find the total resistance

[tex] R_{total}=R_3+\frac{1}{\frac{1}{R_1}+\frac{1}{R_2}} [/tex]

however, this works out different when i derive the resistance of the circuit anew from kirchoffs law which would give

[tex] R_{total}=\frac{1}{\frac{1}{R_1+R_3}+\frac{1}{R_2+R_3}} [/tex]

am i missing something? or is it wrong to apply the resistance formula in the case where there is another resistance connected in series?