Concept:
Compounds in the same homologous series have the same functional group and general formula and differ by a –CH$_2$ unit.
Explanation:
- Alkanes follow the general formula C$_n$H$_{2n+2}$.
- C$_5$H$_{12}$ is an alkane (fits the formula).
- C$_4$H$_8$ follows C$_n$H$_{2n}$, which is the formula of alkenes.
Thus, they belong to different homologous series.
Conclusion:
C$_4$H$_8$ and C$_5$H$_{12}$ do not belong to the same homologous series.