Question

What is the ratio of equivalent resistance of series combination of n equal resistance to the quivalent resistance in parallel combination of three n resistors?

Answer 1

$$\begin{gathered} \mathrm{Let} \\ \mathrm{R}=\mathrm{Resistance}\mathrm{of}\mathrm{each}\mathrm{resistor} \\ \mathrm{Series}\mathrm{combination}\mathrm{of}\mathrm{n}\mathrm{equal}\mathrm{resistors}: \\ {\mathrm{R}}_{\mathrm{s}}=\mathrm{nR} \\ \mathrm{Parallel}\mathrm{combination}\mathrm{of}\mathrm{n}\mathrm{equal}\mathrm{resistors}: \\ {\mathrm{R}}_{\mathrm{p}}=\frac{\mathrm{R}}{\mathrm{n}} \\ \mathrm{Ratio}: \\ \frac{{\mathrm{R}}_{\mathrm{s}}}{{\mathrm{R}}_{\mathrm{p}}}=\frac{\mathrm{nR}}{\frac{\mathrm{R}}{\mathrm{n}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}{\mathbf{n}}^{\mathbf{2}} \end{gathered}$$