The successive ionization energies of an unknown element X are:
1st - 9534; 2nd - 11790; 3rd - 13683; 4th - 15309; 5th - 16458.
To what group number does X belong?
There aren't any significant jumps so that means X is in groups 5-8. After that I don't know.
There are other little jumps to look out for, like the removal of the first electron from a filled subshell. Consider the ionization energies (in eV) for the third period elements:
$$ \begin{array}{|c|c|c|c|c|c|c|c|c|}\hline \text{electron}&\ce{Ar}&\ce{Cl}&\ce{S}&\ce{P}&\ce{Si}&\ce{Al}&\ce{Mg}&\ce{Na}\\\hline \ce{3p}&1521&-&-&-&-&-&-&-\\\hline \ce{3p}&2666&1251&-&-&-&-&-&-\\\hline \ce{3p}&3931&2298&1000&-&-&-&-&-\\\hline \ce{3p}&5771&3822&2252&1012&-&-&-&-\\\hline \ce{3p}&7238&5159&3357&1907&787&-&-&-\\\hline \ce{3p}&8781&6542&4556&2914&1577&578&-&-\\\hline \ce{3s}&11995&9362&7004&4964&3233&1817&738&-\\\hline \ce{3s}&13842&11018&8496&6374&4356&2745&1451&496\\\hline \ce{2p}&40760&33604&27107&21267&16091&11577&7733&4562\\\hline \end{array} $$
Notice how, in addition to the big jump between $\ce{3s}$ and $\ce{2p}$, there is a small jump between $\ce{3p}$ and $\ce{3s}$ for Ar, Cl, S, P, and Si. If you have five ionization energies, and you don't see the large jump, look for the small jump. If you don't see the small jump, what group is X in?
There is another even smaller jump to look for. Half-filled subshells are also slightly more stable than other configurations (except filled subshells). Thus, it should require a slightly higher increase in ionization energy to remove an electron form any configuration with $\ce{np^3}$ than from $\ce{np^2}$ or $\ce{np^4}$. Let's look at the step wise increase in ionization energy (in other words, how much higher is the ionization energy for the second electron than the ionization energy for the first electron and so on?).
$$ \begin{array}{|c|c|c|c|c|c|c|c|c|}\hline \text{electron}&\ce{Ar}&\ce{Cl}&\ce{S}&\ce{P}&\ce{Si}&\ce{Al}&\ce{Mg}&\ce{Na}\\\hline \ce{3p}&-&-&-&-&-&-&-&-\\\hline \ce{3p}&1145&-&-&-&-&-&-&-\\\hline \ce{3p}&1265&1047&-&-&-&-&-&-\\\hline \ce{3p}&1840&1524&1252&-&-&-&-&-\\\hline \ce{3p}&1467&1337&1105&895&-&-&-&-\\\hline \ce{3p}&1543&1383&1199&1007&790&-&-&-\\\hline \ce{3s}&3214&2820&2448&2050&1656&1239&-&-\\\hline \ce{3s}&1847&1656&1492&1410&1123&928&713&-\\\hline \ce{2p}&26918&22586&18611&14893&11735&8832&6252&4066\\\hline \end{array} $$
You can see that the increase in ionization energy for the third electron to be removed from argon is larger than the increase for the second or fourth electrons. Your numbers don't have this little increase in them at all. Have you considered that X might be a transition metal?
