When the sum of an A.P. is known then for
(i) three (ii) four (iii) five terms.

Answer

To solve the problem of finding terms of an Arithmetic Progression (A.P.) given the sum, we can follow these steps for (i) three terms, (ii) four terms, and (iii) five terms.

### Step-by-Step Solution:

#### (i) Finding three terms of an A.P. with sum K:

1. **Define the terms**: Let the three terms of the A.P. be:
- First term: \( a - d \)
- Second term: \( a \)
- Third term: \( a + d \)

2. **Calculate the sum**: The sum \( S \) of these three terms is:
\[ S = (a - d) + a + (a + d) = 3a \]

3. **Set the sum equal to K**: Since the sum is given as \( K \):
\[ 3a = K \]

4. **Solve for \( a \)**:
\[ a = \frac{K}{3} \]

5. **Write the three terms**: The three terms can now be expressed as:
- First term: \( \frac{K}{3} - d \)
- Second term: \( \frac{K}{3} \)
- Third term: \( \frac{K}{3} + d \)

#### (ii) Finding four terms of an A.P. with sum K:

1. **Define the terms**: Let the four terms of the A.P. be:
- First term: \( a - 3d \)
- Second term: \( a - d \)
- Third term: \( a + d \)
- Fourth term: \( a + 3d \)

2. **Calculate the sum**: The sum \( S \) of these four terms is:
\[ S = (a - 3d) + (a - d) + (a + d) + (a + 3d) = 4a \]

3. **Set the sum equal to K**: Since the sum is given as \( K \):
\[ 4a = K \]

4. **Solve for \( a \)**:
\[ a = \frac{K}{4} \]

5. **Write the four terms**: The four terms can now be expressed as:
- First term: \( \frac{K}{4} - 3d \)
- Second term: \( \frac{K}{4} - d \)
- Third term: \( \frac{K}{4} + d \)
- Fourth term: \( \frac{K}{4} + 3d \)

#### (iii) Finding five terms of an A.P. with sum K:

1. **Define the terms**: Let the five terms of the A.P. be:
- First term: \( a - 2d \)
- Second term: \( a - d \)
- Third term: \( a \)
- Fourth term: \( a + d \)
- Fifth term: \( a + 2d \)

2. **Calculate the sum**: The sum \( S \) of these five terms is:
\[ S = (a - 2d) + (a - d) + a + (a + d) + (a + 2d) = 5a \]

3. **Set the sum equal to K**: Since the sum is given as \( K \):
\[ 5a = K \]

4. **Solve for \( a \)**:
\[ a = \frac{K}{5} \]

5. **Write the five terms**: The five terms can now be expressed as:
- First term: \( \frac{K}{5} - 2d \)
- Second term: \( \frac{K}{5} - d \)
- Third term: \( \frac{K}{5} \)
- Fourth term: \( \frac{K}{5} + d \)
- Fifth term: \( \frac{K}{5} + 2d \)

### Summary of Results:
- For three terms: \( \left( \frac{K}{3} - d, \frac{K}{3}, \frac{K}{3} + d \right) \)
- For four terms: \( \left( \frac{K}{4} - 3d, \frac{K}{4} - d, \frac{K}{4} + d, \frac{K}{4} + 3d \right) \)
- For five terms: \( \left( \frac{K}{5} - 2d, \frac{K}{5} - d, \frac{K}{5}, \frac{K}{5} + d, \frac{K}{5} + 2d \right) \)