The perimeter of a square is given by the formula:

$$\text{Perimeter} = 4 \times \text{side}$$

Using this, the perimeters for the given sides are:

• For side $1$ cm: $4 \times 1 =$ $4$ cm
• For side $1.5$ cm: $4 \times 1.5 =$ $6$ cm
• For side $2$ cm: $4 \times 2 =$ $8$ cm
• For side $2.5$ cm: $4 \times 2.5 =$ $10$ cm
• For side $3$ cm: $4 \times 3 =$ $12$ cm

When the side of the square increases by $0.5$ cm, the perimeter increases by $2$ cm each time. This is because the perimeter is a linear polynomial in the side length ($4x$), and an increase of $0.5$ cm in the side results in an increase of $4 \times 0.5 = 2$ cm in the perimeter. Thus, the perimeters form a linear pattern with a constant difference of $2$.