Arrange the following ions in order of their increasing size: `Li^(+), Mg^(2+), K^(+), AI^(3+)`.

Explanation

To arrange the ions \( \text{Li}^+, \text{Mg}^{2+}, \text{K}^+, \text{Al}^{3+} \) in order of their increasing size, we need to consider the following factors:

1. **Charge of the Ions**: The greater the positive charge on the ion, the smaller the size due to increased nuclear attraction on the remaining electrons.
2. **Number of Electrons**: Ions with the same number of electrons but different nuclear charges will have different sizes. The more protons in the nucleus (higher atomic number), the stronger the pull on the electrons, leading to a smaller ionic size.

### Step-by-Step Solution:

1. **Identify the Ions and Their Charges**:
- \( \text{Li}^+ \) (1+ charge)
- \( \text{Mg}^{2+} \) (2+ charge)
- \( \text{K}^+ \) (1+ charge)
- \( \text{Al}^{3+} \) (3+ charge)

2. **Determine the Number of Electrons**:
- \( \text{Li}^+ \) has 2 electrons (Li has 3 electrons, loses 1).
- \( \text{Mg}^{2+} \) has 10 electrons (Mg has 12 electrons, loses 2).
- \( \text{K}^+ \) has 18 electrons (K has 19 electrons, loses 1).
- \( \text{Al}^{3+} \) has 10 electrons (Al has 13 electrons, loses 3).

3. **Analyze the Nuclear Charge**:
- **Lithium**: \( Z = 3 \) (3 protons)
- **Magnesium**: \( Z = 12 \) (12 protons)
- **Potassium**: \( Z = 19 \) (19 protons)
- **Aluminum**: \( Z = 13 \) (13 protons)

4. **Compare the Sizes Based on Charge and Electrons**:
- **Aluminum \( \text{Al}^{3+} \)** has the highest positive charge and 10 electrons, making it the smallest ion.
- **Magnesium \( \text{Mg}^{2+} \)** has 10 electrons as well, but only 12 protons, making it larger than \( \text{Al}^{3+} \).
- **Lithium \( \text{Li}^+ \)** has 2 electrons and 3 protons, making it larger than \( \text{Mg}^{2+} \).
- **Potassium \( \text{K}^+ \)** has 18 electrons and 19 protons, making it the largest ion.

5. **Final Order of Increasing Size**:
- \( \text{Al}^{3+} < \text{Mg}^{2+} < \text{Li}^+ < \text{K}^+ \)

### Conclusion:
The order of increasing size of the ions is:
\[ \text{Al}^{3+} < \text{Mg}^{2+} < \text{Li}^+ < \text{K}^+ \]