Which of the following options is INCORRECT?

Options

A. The number of terms in the A.P, 3, 6, 9, 12,..., 111 is 37.
B. If the first three terms of an A.P. are x-1, x + 1 and 2x + 3, then the value of x is 0.
C. The sum of first 'n' natural numbers is `[(n(n + 1))/(2)]^(2)`
D. None of these

Answer

To solve the question of identifying the incorrect option, we will analyze each option step by step.

### Step 1: Analyze Option 1
**Statement**: The number of terms in the AP (Arithmetic Progression) 3, 6, 9, ..., 111 is 37.

**Solution**:
1. Identify the first term \( a = 3 \) and the last term \( l = 111 \).
2. The common difference \( d = 6 - 3 = 3 \).
3. Use the formula for the \( n \)-th term of an AP:
\[ a_n = a + (n - 1) \cdot d \]
Setting \( a_n = 111 \):
\[ 111 = 3 + (n - 1) \cdot 3 \]
4. Rearranging gives:
\[ 111 - 3 = (n - 1) \cdot 3 \implies 108 = (n - 1) \cdot 3 \]
5. Dividing both sides by 3:
\[ n - 1 = 36 \implies n = 37 \]
**Conclusion**: Option 1 is correct.

### Step 2: Analyze Option 2
**Statement**: The first three terms of an AP are \( x - 1, x + 1, 2x + 3 \). Then, the value of \( x \) is 0.

**Solution**:
1. For the terms to be in AP, the middle term must be the average of the other two:
\[ x + 1 = \frac{(x - 1) + (2x + 3)}{2} \]
2. Simplifying the right side:
\[ x + 1 = \frac{x - 1 + 2x + 3}{2} = \frac{3x + 2}{2} \]
3. Cross-multiply:
\[ 2(x + 1) = 3x + 2 \implies 2x + 2 = 3x + 2 \]
4. Rearranging gives:
\[ 2 = 3x - 2x \implies x = 0 \]
**Conclusion**: Option 2 is correct.

### Step 3: Analyze Option 3
**Statement**: The sum of the first \( n \) natural numbers is \( \frac{n(n + 1)}{2} = 2 \).

**Solution**:
1. The formula for the sum of the first \( n \) natural numbers is indeed \( \frac{n(n + 1)}{2} \).
2. Setting this equal to 2:
\[ \frac{n(n + 1)}{2} = 2 \]
3. Multiplying both sides by 2:
\[ n(n + 1) = 4 \]
4. Rearranging gives:
\[ n^2 + n - 4 = 0 \]
5. Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ n = \frac{-1 \pm \sqrt{1 + 16}}{2} = \frac{-1 \pm \sqrt{17}}{2} \]
6. Since \( n \) must be a natural number, there are no valid solutions for \( n \) in this case.
**Conclusion**: Option 3 is incorrect.

### Final Answer
The incorrect option is **Option 3**.