If the sum of the first `11` terms of an arithmetical progression equals that of the first `19` terms, then the sum of its first `30` terms, is (A) equal to 0 (B) equal to -1 (C) equal to 1 (D) non unique

Options

A. equal to 0
B. equal to `-1`
C. equal to 1
D. non unique

Answer

To solve the problem, we need to find the sum of the first 30 terms of an arithmetic progression (AP) given that the sum of the first 11 terms is equal to the sum of the first 19 terms.

### Step-by-Step Solution:

1. **Understanding the Formula for the Sum of an AP**:
The sum of the first \( n \) terms of an arithmetic progression is given by the formula:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
where \( a \) is the first term and \( d \) is the common difference.

2. **Finding the Sum of the First 11 Terms**:
Using the formula for \( n = 11 \):
\[ S_{11} = \frac{11}{2} \times (2a + (11 - 1)d) = \frac{11}{2} \times (2a + 10d) \]
Simplifying this, we get:
\[ S_{11} = 11(a + 5d) \]

3. **Finding the Sum of the First 19 Terms**:
Now, for \( n = 19 \):
\[ S_{19} = \frac{19}{2} \times (2a + (19 - 1)d) = \frac{19}{2} \times (2a + 18d) \]
Simplifying this, we have:
\[ S_{19} = 19(a + 9d) \]

4. **Setting the Two Sums Equal**:
According to the problem, \( S_{11} = S_{19} \):
\[ 11(a + 5d) = 19(a + 9d) \]

5. **Expanding and Rearranging the Equation**:
Expanding both sides gives:
\[ 11a + 55d = 19a + 171d \]
Rearranging this, we get:
\[ 11a - 19a + 55d - 171d = 0 \]
Simplifying further:
\[ -8a - 116d = 0 \]
Dividing the entire equation by 4:
\[ -2a - 29d = 0 \quad \Rightarrow \quad 2a + 29d = 0 \quad \text{(Equation 1)} \]

6. **Finding the Sum of the First 30 Terms**:
Now, we need to find \( S_{30} \):
\[ S_{30} = \frac{30}{2} \times (2a + (30 - 1)d) = 15 \times (2a + 29d) \]
From Equation 1, we know that \( 2a + 29d = 0 \):
\[ S_{30} = 15 \times 0 = 0 \]

### Conclusion:
The sum of the first 30 terms of the arithmetic progression is \( 0 \).

### Final Answer:
The correct option is (A) equal to 0.