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. 3
C. 5
D. 6

Answer

To determine which of the given numbers can be expressed in the form of \( n(n + 1)(n + 2) \), where \( n \) is a natural number, we will evaluate each number one by one.

### Step 1: Understand the Expression
The expression \( n(n + 1)(n + 2) \) represents the product of three consecutive natural numbers starting from \( n \).

### Step 2: Evaluate Each Option
We will check each number (7, 3, 5, and 6) to see if it can be expressed in the form of \( n(n + 1)(n + 2) \).

#### Option 1: Check for 7
- We need to find \( n \) such that \( n(n + 1)(n + 2) = 7 \).
- Testing \( n = 1 \):
\[ 1(1 + 1)(1 + 2) = 1 \cdot 2 \cdot 3 = 6 \quad \text{(not 7)} \]
- Testing \( n = 2 \):
\[ 2(2 + 1)(2 + 2) = 2 \cdot 3 \cdot 4 = 24 \quad \text{(not 7)} \]
- Conclusion: 7 cannot be expressed in the form.

#### Option 2: Check for 3
- We need to find \( n \) such that \( n(n + 1)(n + 2) = 3 \).
- Testing \( n = 1 \):
\[ 1(1 + 1)(1 + 2) = 1 \cdot 2 \cdot 3 = 6 \quad \text{(not 3)} \]
- Testing \( n = 0 \):
\[ 0(0 + 1)(0 + 2) = 0 \cdot 1 \cdot 2 = 0 \quad \text{(not 3)} \]
- Conclusion: 3 cannot be expressed in the form.

#### Option 3: Check for 5
- We need to find \( n \) such that \( n(n + 1)(n + 2) = 5 \).
- Testing \( n = 1 \):
\[ 1(1 + 1)(1 + 2) = 1 \cdot 2 \cdot 3 = 6 \quad \text{(not 5)} \]
- Testing \( n = 0 \):
\[ 0(0 + 1)(0 + 2) = 0 \cdot 1 \cdot 2 = 0 \quad \text{(not 5)} \]
- Conclusion: 5 cannot be expressed in the form.

#### Option 4: Check for 6
- We need to find \( n \) such that \( n(n + 1)(n + 2) = 6 \).
- Testing \( n = 1 \):
\[ 1(1 + 1)(1 + 2) = 1 \cdot 2 \cdot 3 = 6 \quad \text{(matches)} \]
- Conclusion: 6 can be expressed in the form.

### Final Conclusion
Among the options provided, the only number that can be expressed in the form \( n(n + 1)(n + 2) \) is **6**.