The correct order of increasing radii are

Options

A. `Be^(2+), Mg^(2+), Na^(o+)`
B. `K^(o+), Ca^(2+), S^(2-)`
C. `O^(2-), F^(ɵ), N^(3-)`
D. `S^(2-), O^(2-), As^(3-)`

Explanation

To determine the correct order of increasing radii among the given ions, we need to analyze each option based on the principles of ionic size and isoelectronic species.

### Step-by-Step Solution:

1. **Understanding Ionic Sizes**:
- Cations (positively charged ions) are smaller than their parent atoms because they lose electrons, leading to a decrease in electron-electron repulsion.
- Anions (negatively charged ions) are larger than their parent atoms because they gain electrons, increasing electron-electron repulsion.

2. **Analyzing Option A**:
- The ions given are \( \text{Be}^{2+}, \text{Mg}^{2+}, \text{Na}^{+} \).
- \( \text{Be}^{2+} \) has the smallest radius because it has the highest positive charge (+2), which pulls the electrons closer to the nucleus.
- \( \text{Mg}^{2+} \) is larger than \( \text{Be}^{2+} \) but smaller than \( \text{Na}^{+} \) because it also has a +2 charge.
- \( \text{Na}^{+} \) is the largest as it has a +1 charge.
- Thus, the order is:
\[ \text{Be}^{2+} < \text{Mg}^{2+} < \text{Na}^{+} \]

3. **Analyzing Option B**:
- The ions given are \( \text{K}^{+}, \text{Ca}^{2+}, \text{S}^{2-} \).
- All these ions are isoelectronic with the electronic configuration of Argon.
- For isoelectronic species, the size of cations decreases with increasing positive charge, while the size of anions increases with increasing negative charge.
- Thus, the order is:
\[ \text{Ca}^{2+} < \text{K}^{+} < \text{S}^{2-} \]
- This option is incorrect as it does not follow the increasing order.

4. **Analyzing Option C**:
- The ions given are \( \text{O}^{2-}, \text{F}^{-}, \text{N}^{3-} \).
- These ions are also isoelectronic with the electronic configuration of Neon.
- The order of increasing size for anions is:
\[ \text{F}^{-} < \text{O}^{2-} < \text{N}^{3-} \]
- This option is also incorrect.

5. **Analyzing Option D**:
- The ions given are \( \text{S}^{2-}, \text{O}^{2-}, \text{As}^{3-} \).
- Here, \( \text{S}^{2-} \) is larger than \( \text{O}^{2-} \) due to the increased number of electrons.
- \( \text{As}^{3-} \) will be the largest due to its position in the periodic table and the additional negative charge.
- Thus, the order is:
\[ \text{O}^{2-} < \text{S}^{2-} < \text{As}^{3-} \]
- This option is also incorrect.

### Conclusion:
The correct order of increasing radii is found in **Option A**:
\[ \text{Be}^{2+} < \text{Mg}^{2+} < \text{Na}^{+} \]