If the sum of first 11 terms of an arithmetic progression equal that of the first 19 terms, Then, what si the sum of first 30 terms?

Options

A. 0
B. -1
C. 1
D. Not 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. **Understand the formula for the sum of the first n terms of an AP**:
The sum \( S_n \) of the first \( n \) terms of an arithmetic progression can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the number of terms.

2. **Set up the equations for the sums**:
- For the first 11 terms:
\[ S_{11} = \frac{11}{2} \times (2a + 10d) \]
- For the first 19 terms:
\[ S_{19} = \frac{19}{2} \times (2a + 18d) \]

3. **Equate the two sums**:
According to the problem, the sum of the first 11 terms is equal to the sum of the first 19 terms:
\[ \frac{11}{2} \times (2a + 10d) = \frac{19}{2} \times (2a + 18d) \]

4. **Simplify the equation**:
We can eliminate \(\frac{1}{2}\) from both sides:
\[ 11(2a + 10d) = 19(2a + 18d) \]
Expanding both sides:
\[ 22a + 110d = 38a + 342d \]

5. **Rearranging the equation**:
Bringing like terms together:
\[ 22a - 38a = 342d - 110d \]
This simplifies to:
\[ -16a = 232d \]
Dividing both sides by -16 gives:
\[ a = -\frac{232}{16}d = -\frac{29}{2}d \]

6. **Find the sum of the first 30 terms**:
Now, we can find the sum of the first 30 terms:
\[ S_{30} = \frac{30}{2} \times (2a + 29d) = 15 \times (2a + 29d) \]
Substitute \( a = -\frac{29}{2}d \):
\[ S_{30} = 15 \times \left(2\left(-\frac{29}{2}d\right) + 29d\right) \]
Simplifying:
\[ S_{30} = 15 \times \left(-29d + 29d\right) = 15 \times 0 = 0 \]

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

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