Resistance of the Network: The resistor network consists of: One resistor \(R\) in series with two resistors \(R\) and \(R\) in parallel. Step 1: Equivalent resistance of the parallel resistors The formula for the equivalent resistance \(R_{\text{parallel}}\) of two resistors \(R_1\) and \(R_2\) in parallel is: \[\frac{1}{R_{\text{parallel}}} = \frac{1}{R_1} + \frac{1}{R_2}\] Here, \(R_1 = R\) and \(R_2 = R\). Therefore: \[\frac{1}{R_{\text{parallel}}} = \frac{1}{R} + \frac{1}{R} = \frac{2}{R}\] \[R_{\text{parallel}} = \frac{R}{2}\] Step 2: Total resistance of the network The equivalent resistance \(R_{\text{total}}\) is the series combination of the first resistor \(R\) and the parallel equivalent resistance \(R_{\text{parallel}}\): \[R_{\text{total}} = R + R_{\text{parallel}}\] Substitute \(R_{\text{parallel}} = \frac{R}{2}\): \[R_{\text{total}} = R + \frac{R}{2}\] \[R_{\text{total}} = \frac{2R}{2} + \frac{R}{2} = \frac{3R}{2}\] Final Answer: The total resistance of the network is: \[R_{\text{total}} = \frac{3R}{2}.\]
