Write four more rational number in each of the patterns:
`(-2)/(3), (2)/(-3) , (4)/(-6), (6)/(-9),...`

Answer

To find four more rational numbers in the given pattern, we can observe the sequence and identify the pattern. The first term is \(-\frac{2}{3}\), and the subsequent terms can be derived by multiplying the previous term by \(-\frac{n}{n}\), where \(n\) is a positive integer that increases with each step.

Let's break this down step by step:

### Step 1: Identify the first term
The first term is:
\[ -\frac{2}{3} \]

### Step 2: Multiply by \(-\frac{1}{1}\) to get the second term
\[ -\frac{2}{3} \times -\frac{1}{1} = \frac{2}{3} \]

### Step 3: Multiply by \(-\frac{2}{2}\) to get the third term
\[ -\frac{2}{3} \times -\frac{2}{2} = \frac{4}{-6} = -\frac{4}{6} = -\frac{2}{3} \]

### Step 4: Multiply by \(-\frac{3}{3}\) to get the fourth term
\[ -\frac{2}{3} \times -\frac{3}{3} = \frac{6}{-9} = -\frac{6}{9} = -\frac{2}{3} \]

### Step 5: Continue the pattern
Now we will continue this pattern by multiplying by \(-\frac{4}{4}\), \(-\frac{5}{5}\), \(-\frac{6}{6}\), and \(-\frac{7}{7}\).

#### Fifth term:
\[ -\frac{2}{3} \times -\frac{4}{4} = \frac{8}{-12} = -\frac{8}{12} = -\frac{2}{3} \]

#### Sixth term:
\[ -\frac{2}{3} \times -\frac{5}{5} = \frac{10}{-15} = -\frac{10}{15} = -\frac{2}{3} \]

#### Seventh term:
\[ -\frac{2}{3} \times -\frac{6}{6} = \frac{12}{-18} = -\frac{12}{18} = -\frac{2}{3} \]

#### Eighth term:
\[ -\frac{2}{3} \times -\frac{7}{7} = \frac{14}{-21} = -\frac{14}{21} = -\frac{2}{3} \]

### Final Answer:
The four additional rational numbers in the pattern are:
1. \(-\frac{8}{12}\)
2. \(-\frac{10}{15}\)
3. \(-\frac{12}{18}\)
4. \(-\frac{14}{21}\)