Obtain the expression for the equivalent resistance for $3$ resistors connected in parallel and also write the expression of equivalent resistance for connection of $n$ resistors.

(N/A) When connecting the terminals of a battery of voltage $V$ to points $a$ and $b$,a total current $I$ flows through the circuit. The currents through resistors $R_{1}, R_{2}, R_{3}$ are $I_{1}, I_{2}, I_{3}$ respectively. According to Ohm's law,the potential difference across each resistor in parallel is the same,equal to $V$.
$\therefore V = I_{1} R_{1} \Rightarrow I_{1} = \frac{V}{R_{1}} \quad \dots (1)$
$V = I_{2} R_{2} \Rightarrow I_{2} = \frac{V}{R_{2}} \quad \dots (2)$
$V = I_{3} R_{3} \Rightarrow I_{3} = \frac{V}{R_{3}} \quad \dots (3)$
At junction $a$,the total current is the sum of individual currents:
$I = I_{1} + I_{2} + I_{3} \quad \dots (4)$
Substituting equations $(1), (2),$ and $(3)$ into equation $(4)$:
$I = \frac{V}{R_{1}} + \frac{V}{R_{2}} + \frac{V}{R_{3}}$
Dividing both sides by $V$:
$\frac{I}{V} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}$
If the equivalent resistance is $R_{p}$,then by Ohm's law $I = \frac{V}{R_{p}}$,so $\frac{I}{V} = \frac{1}{R_{p}}$.
Therefore,for $3$ resistors in parallel:
$\frac{1}{R_{p}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}$
For $n$ resistors connected in parallel,the expression is:
$\frac{1}{R_{p}} = \sum_{i=1}^{n} \frac{1}{R_{i}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \dots + \frac{1}{R_{n}}$