Final answer:
Judging by the large increase in ionization energy after the third electron is removed, it indicates that the unknown element likely belongs to Group 3 of the periodic table.
Step-by-step explanation:
To determine which family the unknown element with the given successive ionization energies belongs to, we must look at the pattern of the ionization energies. A significant jump between successive ionization energies typically indicates the removal of an electron from a new, closer to the nucleus energy level, signifying that the previous energy level has been emptied. This usually happens after the valence electrons have been removed.
In the given data, there's a substantial increase from the third ionization energy (9144 kJ/mol) to the fourth (18724 kJ/mol). This suggests that the first three electrons are much easier to remove and are likely the valence electrons. Thus, the element is likely from Group 3 on the periodic table because it appears to have three valence electrons.
An element from Group 3 would exhibit such a pattern of ionization energies, indicative of having three electrons in its outermost shell.