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If you think about what n, represents, it is the shell of an atom. When the shell is increasing, there are now a great number of electrons (or places where electrons could be). In a neutral atom, the number of protons and electrons are the same so this increase in electrons correlates to an increase in protons, therefore making the nucleus of the atom larger. With a larger nucleus and electron "cloud", the atom will be bigger.
The reduced electrostatic attraction due to the distance from the nucleus, electron-electron repulsions, and shielding from the inner energy levels and subshells results in an effective nuclear charge, and the result is an increase in the size of the atom.
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