The number which can be written in the form of n (n+1) (n+ 2), where n is a natural number is

Options

A. 7
B. 5
C. 3
D. 6

Answer

To solve the problem, we need to find a number that can be expressed in the form of \( n(n+1)(n+2) \), where \( n \) is a natural number. Let's go through the steps systematically.

### Step 1: Understanding the Expression
The expression \( n(n+1)(n+2) \) represents the product of three consecutive natural numbers. For any natural number \( n \), \( n \), \( n+1 \), and \( n+2 \) are three consecutive integers.

### Step 2: Evaluating the Given Options
We need to check each of the provided options (7, 5, 3, and 6) to see if they can be expressed in the form \( n(n+1)(n+2) \).

### Step 3: Testing Each Option
1. **Option 7**:
- Check if \( n(n+1)(n+2) = 7 \) for any natural number \( n \).
- Testing \( n = 1 \): \( 1 \times 2 \times 3 = 6 \)
- Testing \( n = 2 \): \( 2 \times 3 \times 4 = 24 \)
- 7 cannot be expressed in this form.

2. **Option 5**:
- Check if \( n(n+1)(n+2) = 5 \).
- Testing \( n = 1 \): \( 1 \times 2 \times 3 = 6 \)
- Testing \( n = 2 \): \( 2 \times 3 \times 4 = 24 \)
- 5 cannot be expressed in this form.

3. **Option 3**:
- Check if \( n(n+1)(n+2) = 3 \).
- Testing \( n = 1 \): \( 1 \times 2 \times 3 = 6 \)
- Testing \( n = 0 \): \( 0 \times 1 \times 2 = 0 \)
- 3 cannot be expressed in this form.

4. **Option 6**:
- Check if \( n(n+1)(n+2) = 6 \).
- Testing \( n = 1 \): \( 1 \times 2 \times 3 = 6 \)
- Here, \( n = 1 \) works perfectly.
- Therefore, 6 can be expressed as \( 1(1+1)(1+2) = 6 \).

### Conclusion
The only number from the options that can be expressed in the form \( n(n+1)(n+2) \) is **6**.

### Final Answer
**6**