If the sum of first 20 terms of an A.P. is same as the sum of its first 28 terms, find the sum of its 48 terms.

Answer

To solve the problem, we need to find the sum of the first 48 terms of an arithmetic progression (A.P.) given that the sum of the first 20 terms is equal to the sum of the first 28 terms.

### Step-by-step Solution:

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

2. **Finding the Sum of the First 20 Terms**:
For the first 20 terms:
\[ S_{20} = \frac{20}{2} \left(2a + (20 - 1)d\right) = 10(2a + 19d) \]
Let's denote this as Equation (1):
\[ S_{20} = 10(2a + 19d) \]

3. **Finding the Sum of the First 28 Terms**:
For the first 28 terms:
\[ S_{28} = \frac{28}{2} \left(2a + (28 - 1)d\right) = 14(2a + 27d) \]
Let's denote this as Equation (2):
\[ S_{28} = 14(2a + 27d) \]

4. **Setting the Two Sums Equal**:
According to the problem, the sum of the first 20 terms is equal to the sum of the first 28 terms:
\[ 10(2a + 19d) = 14(2a + 27d) \]

5. **Expanding and Rearranging the Equation**:
Expanding both sides:
\[ 20a + 190d = 28a + 378d \]
Rearranging gives:
\[ 20a - 28a = 378d - 190d \]
Simplifying:
\[ -8a = 188d \]
Thus, we can express \( a \) in terms of \( d \):
\[ a = -\frac{188}{8}d = -\frac{47}{2}d \]

6. **Finding the Sum of the First 48 Terms**:
Now we need to calculate the sum of the first 48 terms:
\[ S_{48} = \frac{48}{2} \left(2a + (48 - 1)d\right) = 24(2a + 47d) \]
Substituting \( a = -\frac{47}{2}d \):
\[ S_{48} = 24\left(2\left(-\frac{47}{2}d\right) + 47d\right) \]
Simplifying:
\[ S_{48} = 24\left(-47d + 47d\right) = 24(0) = 0 \]

### Final Answer:
The sum of the first 48 terms of the A.P. is **0**.