If the sum of the first 9 terms of an AP is equal to the sum of its first 11 terms, then find the sum of its first 20 terms.

Answer

To solve the problem step-by-step, we will use the formula for the sum of the first n terms of an arithmetic progression (AP):

The formula for the sum of the first n terms of an AP is given by:
\[ S_n = \frac{n}{2} \times (2A + (n - 1)D) \]
where:
- \( S_n \) = sum of the first n terms
- \( A \) = first term of the AP
- \( D \) = common difference
- \( n \) = number of terms

### Step 1: Set up the equations for \( S_9 \) and \( S_{11} \)
We know from the problem that:
\[ S_9 = S_{11} \]

Using the formula for the sum of the first n terms:
\[ S_9 = \frac{9}{2} \times (2A + (9 - 1)D) = \frac{9}{2} \times (2A + 8D) \]
\[ S_{11} = \frac{11}{2} \times (2A + (11 - 1)D) = \frac{11}{2} \times (2A + 10D) \]

### Step 2: Set the two equations equal to each other
Since \( S_9 = S_{11} \), we can set the two equations equal:
\[ \frac{9}{2} \times (2A + 8D) = \frac{11}{2} \times (2A + 10D) \]

### Step 3: Eliminate the fractions
Multiply both sides by 2 to eliminate the fractions:
\[ 9(2A + 8D) = 11(2A + 10D) \]

### Step 4: Expand both sides
Expanding both sides gives:
\[ 18A + 72D = 22A + 110D \]

### Step 5: Rearrange the equation
Rearranging the equation to isolate terms involving \( A \) and \( D \):
\[ 18A - 22A = 110D - 72D \]
\[ -4A = 38D \]

### Step 6: Solve for \( A \) in terms of \( D \)
Dividing both sides by -4 gives:
\[ A = -\frac{38}{4}D = -\frac{19}{2}D \]

### Step 7: Find \( S_{20} \)
Now, we need to find the sum of the first 20 terms \( S_{20} \):
\[ S_{20} = \frac{20}{2} \times (2A + (20 - 1)D) \]
\[ S_{20} = 10 \times (2A + 19D) \]

### Step 8: Substitute \( A \) into the equation
Substituting \( A = -\frac{19}{2}D \) into the equation:
\[ S_{20} = 10 \times \left(2 \left(-\frac{19}{2}D\right) + 19D\right) \]
\[ S_{20} = 10 \times \left(-19D + 19D\right) \]
\[ S_{20} = 10 \times 0 \]
\[ S_{20} = 0 \]

### Final Answer
The sum of the first 20 terms of the AP is:
\[ S_{20} = 0 \]

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