Write four more rational number in each of the patterns:
`(-1)/(4), (-2)/(8), (-3)/(12),`

Answer

To find four more rational numbers in the given pattern of rational numbers \((-1/4), (-2/8), (-3/12)\), we will first identify the pattern in the numerators and denominators.

### Step 1: Identify the pattern in the numerators
The numerators of the given rational numbers are:
- For \(-1/4\), the numerator is \(-1\)
- For \(-2/8\), the numerator is \(-2\)
- For \(-3/12\), the numerator is \(-3\)

We can see that the numerators are \(-n\) where \(n\) starts from 1 and increases by 1 for each subsequent term.

### Step 2: Identify the pattern in the denominators
The denominators of the given rational numbers are:
- For \(-1/4\), the denominator is \(4\)
- For \(-2/8\), the denominator is \(8\)
- For \(-3/12\), the denominator is \(12\)

We can observe that the denominators can be expressed as \(4n\) where \(n\) is the same value used in the numerators.

### Step 3: Write the general formula
From the observations:
- The general form for the \(n^{th}\) term is given by:
\[ \text{Rational Number} = \frac{-n}{4n} \]

### Step 4: Calculate the next four rational numbers
Now we will calculate the next four rational numbers by substituting \(n = 4, 5, 6, 7\):

1. For \(n = 4\):
\[ \frac{-4}{4 \times 4} = \frac{-4}{16} \]

2. For \(n = 5\):
\[ \frac{-5}{4 \times 5} = \frac{-5}{20} \]

3. For \(n = 6\):
\[ \frac{-6}{4 \times 6} = \frac{-6}{24} \]

4. For \(n = 7\):
\[ \frac{-7}{4 \times 7} = \frac{-7}{28} \]

### Final Answer
The next four rational numbers in the pattern are:
\[ \frac{-4}{16}, \frac{-5}{20}, \frac{-6}{24}, \frac{-7}{28} \]