Write four rational numbers in the following pattern `-(1)/(6),(2)/(-12),(3)/(-18),(4)/(-24)`, ________,

Options

A. `(5)/(6),7/8,9/10,11/12`
B. `(5)/(30),(6)/(36),(7)/(42),(8)/(48)`
C. `(5)/(-6),(6)/(-12),(7)/(-18),(8)/(-24)`
D. `(5)/(-30),(6)/(-36),(7)/(-42),(8)/(-48)`

Answer

To find four rational numbers in the given pattern `-(1)/(6), (2)/(-12), (3)/(-18), (4)/(-24)`, we will analyze the pattern in the numerators and denominators.

### Step 1: Identify the pattern in the numerators
The numerators in the given rational numbers are:
- For `-(1)/(6)`, the numerator is `-1`
- For `(2)/(-12)`, the numerator is `2`
- For `(3)/(-18)`, the numerator is `3`
- For `(4)/(-24)`, the numerator is `4`

The numerators are increasing by 1 each time, starting from `-1`.

### Step 2: Identify the pattern in the denominators
The denominators in the given rational numbers are:
- For `-(1)/(6)`, the denominator is `6`
- For `(2)/(-12)`, the denominator is `-12`
- For `(3)/(-18)`, the denominator is `-18`
- For `(4)/(-24)`, the denominator is `-24`

The denominators can be expressed as multiples of `6`:
- `6 = 6 * 1`
- `-12 = 6 * -2`
- `-18 = 6 * -3`
- `-24 = 6 * -4`

### Step 3: Write the next four rational numbers
Following the identified patterns:
1. The next numerator after `4` would be `5`.
2. The next denominator would be `-30` (which is `6 * -5`).

So the next rational number would be `5/(-30)`.

Continuing this:
- The next numerator after `5` would be `6`, and the denominator would be `-36` (which is `6 * -6`), giving us `6/(-36)`.
- The next numerator after `6` would be `7`, and the denominator would be `-42` (which is `6 * -7`), giving us `7/(-42)`.
- The next numerator after `7` would be `8`, and the denominator would be `-48` (which is `6 * -8`), giving us `8/(-48)`.

### Final Answer
The four rational numbers in the pattern are:
- `5/(-30)`
- `6/(-36)`
- `7/(-42)`
- `8/(-48)`