The sum of first 5 terms and first 15 terms of an A.P. are equal. Find the sum of its first 20 terms.

Answer

To solve the problem step by step, we will follow the given information and apply the formula for the sum of an arithmetic progression (A.P.).

### Step 1: Understand the Problem
We know that the sum of the first 5 terms (S5) and the sum of the first 15 terms (S15) of an A.P. are equal. We need to find the sum of the first 20 terms (S20).

### Step 2: Write the Formula for the Sum of n Terms of an A.P.
The formula for the sum of the first n terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
where:
- \( S_n \) is the sum of the first n terms,
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the number of terms.

### Step 3: Set Up the Equations
Since we know that \( S_5 = S_{15} \), we can write:
\[ S_5 = \frac{5}{2} \times (2a + (5 - 1)d) = \frac{5}{2} \times (2a + 4d) \]
\[ S_{15} = \frac{15}{2} \times (2a + (15 - 1)d) = \frac{15}{2} \times (2a + 14d) \]
Setting these two equal gives us:
\[ \frac{5}{2} \times (2a + 4d) = \frac{15}{2} \times (2a + 14d) \]

### Step 4: Simplify the Equation
We can eliminate \(\frac{1}{2}\) from both sides:
\[ 5(2a + 4d) = 15(2a + 14d) \]
Expanding both sides:
\[ 10a + 20d = 30a + 210d \]

### Step 5: Rearrange the Equation
Rearranging gives:
\[ 10a + 20d - 30a - 210d = 0 \]
\[ -20a - 190d = 0 \]
This simplifies to:
\[ 20a = -190d \]
\[ a = -\frac{19}{2}d \]

### Step 6: Find the Sum of the First 20 Terms
Now we will find \( S_{20} \):
\[ S_{20} = \frac{20}{2} \times (2a + (20 - 1)d) = 10 \times (2a + 19d) \]
Substituting \( a = -\frac{19}{2}d \):
\[ S_{20} = 10 \times \left(2 \left(-\frac{19}{2}d\right) + 19d\right) \]
This simplifies to:
\[ S_{20} = 10 \times (-19d + 19d) = 10 \times 0 = 0 \]

### Final Answer
The sum of the first 20 terms of the A.P. is:
\[ \boxed{0} \]