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Anions are ions that have gained an electron from the ground state. The number of protons stay the same, but the electrons increase. As a result the nuclear charge decreases and the electrons are not as tightly bound to the nucleus as before, thus increasing the ionic radii. For cations electrons are lost so we have the same nuclear charge but the electrons are more tightly bound to the nucleus reducing the ionic radii.
The more electrons you have in an atom, the more negative charges you will have and it will result in more negative to negative interactions. If you think of a generic bar magnet, opposite charges (+/-) attract while similar charges (-/-) repel. The same applies to these anions. The more of a negative charge you have, the more e- REPULSIONS you'll have (aka shielding) and thus the radii will increase because more electrons will push the other electrons further from the nucleus.
Anions are larger because they add another electron, creating more electron-electron repulsion between the valence shell electrons and the electrons in the shell below it (increasing distance between the shells). Cations are smaller because they remove an electron, decreasing the amount of electron-electron repulsion between shells.
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