Ok, so because F^+ is already had one of its electron removed which puts it at a higher state than F. If you look at the electronegativity trends, C is smaller than N so requires less ionization energy. That's why I was thinking because F^+ has fewer electons which means it would also requires lower ionization energy. Thus, I feel like i'm dealing with a twisted concept here. So which would you consider bigger between C^2+ and Ne? I'm still confuse. Sorry to bother but this is killing me. x_x
I don't consider electronegativity at all when I consider these problems. I think of the pull of the protons on the electrons. As you move across a row, you get more protons pulling in the electrons so they have a stronger pull and the radius gets smaller. Because the electrons are more tightly held, it is harder to remove one so the ionization energy increases as you move across the row too. If you try to remove a second electron, it is even more tightly held so it requires more energy than taking off the first.
When you go down a column, it works differently. There are more protons so the pull is stronger but the electrons in the valence shell are farther away and the other shells shield them from the charge. Thus it is easier to remove an electron from a element farther down the row.
For the problem, you asked I would think that C^2+ is going to have a higher ionization energy than Ne. With the carbon cation, six protons are pulling on just four electrons so they are going to hold them very tightly. It will require a lot of energy to remove the electron. With Ne, you have ten protons pulling on ten electrons so they aren't held as tightly.
I know that there is more going on than just the pull of the protons on the electrons. The atoms are also have higher ionization energy if you are removing an electron with a full octet. For example, Ne has a higher ionization energy than Mg^+ even though for Ne, it is the first ionization energy and for Mg^+, it is the second ionization energy because in the first case you are breaking the full octet and in the second case you are creating one. However, I don't think the PCAT goes into that level of detail. I think you will be okay if you picture how much the protons are pulling on the electrons. But maybe a chem major can come in with a better explanation.