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Actually, when we add electrons, the attractive pull between the nucleus and electrons are weaker. With more electrons to pull in, the force is weaker. There is also added repulsion between the electrons that cause them to be further apart, creating a larger radius.
An anion has a larger radius than its neutral atom because when you add an electron, the repulsive forces between the electrons is relatively greater than the pull of the positively-charged nucleus, thus causing the electrons to further apart, making the anion's radius larger.
In addition to the comments above, the atomic radii decreases as you move towards the right of a period. This is because as you move right, the added electrons are added to the same n level but the number of protons in the nucleus increases. The protons have a much greater effect than the added electrons, so the protons actually have more pull on the electrons towards the nucleus.
As the people above said, cations are larger than the neutral atoms because there are more electrons that are pushing against each other and expanding the radios. The atomic radius of neutral elements decreases across a period because there are also increasing amounts of protons pulling in the electrons more and more. With a cation, an electron is added but no proton, so it doesn't become smaller.
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