To arrange ions in order of increasing ionic size, we need to consider two main factors: the nuclear charge of the ions and the number of electrons each ion possesses. A general rule is that within an isoelectronic series (ions having the same number of electrons), an ion with a higher nuclear charge will be smaller because of the stronger attraction between the nucleus and the electron cloud.

First, identify the total number of electrons in each species for comparison:

• N^3- would have 7 (number of protons in N) + 3 = 10 electrons.
• Na⁺ would have 11 (number of protons in Na) - 1 = 10 electrons.
• F^- would have 9 (number of protons in F) + 1 = 10 electrons.
• O^2- would have 8 (number of protons in O) + 2 = 10 electrons.
• Mg²⁺ would have 12 (number of protons in Mg) - 2 = 10 electrons.

All these ions are isoelectronic, having the same number of electrons, but the nuclear charge varies, affecting their sizes. The ionic radius decreases typically with an increase in nuclear charge due to the increased pull on the electrons:

• N^3- (10 protons, most negative, least nuclear pull relative to its electron cloud)
• O^2- (8 protons)
• F^- (9 protons)
• Na⁺ (11 protons)
• Mg²⁺ (12 protons, most positive, strongest nuclear pull)

Thus, in increasing order of ionic size, the sequence is:

$$\text{Mg}^{2+} < \text{Na}^+ < \text{F}^- < \text{O}^{2-} < \text{N}^{3-}$$

This arrangement is due to the increasing effect of the nuclear charge in pulling the electron cloud closer as the charge becomes more positive, and the increasing number of electrons or increased negative charge would generally cause an increase in the ionic radius.