Write of the following are A.P.s ? If they form an A.P., find the common difference and write three more terms .
a,2a,3a,4a

Answer

To determine if the sequence \( a, 2a, 3a, 4a \) forms an Arithmetic Progression (A.P.), we need to check if the difference between consecutive terms is constant.

### Step-by-step Solution:

1. **Identify the Terms**: The given terms are:
- First term (\( a_1 \)) = \( a \)
- Second term (\( a_2 \)) = \( 2a \)
- Third term (\( a_3 \)) = \( 3a \)
- Fourth term (\( a_4 \)) = \( 4a \)

2. **Calculate the Common Differences**:
- The difference between the second and first term:
\[ d_1 = a_2 - a_1 = 2a - a = a \]
- The difference between the third and second term:
\[ d_2 = a_3 - a_2 = 3a - 2a = a \]
- The difference between the fourth and third term:
\[ d_3 = a_4 - a_3 = 4a - 3a = a \]

3. **Check if the Common Differences are Equal**:
Since \( d_1 = d_2 = d_3 = a \), the common difference is constant.

4. **Conclusion**: The sequence \( a, 2a, 3a, 4a \) is an Arithmetic Progression (A.P.) with a common difference of \( a \).

5. **Finding Three More Terms**:
To find the next three terms in the A.P., we can continue adding the common difference \( a \):
- Fifth term (\( a_5 \)):
\[ a_5 = a_4 + d = 4a + a = 5a \]
- Sixth term (\( a_6 \)):
\[ a_6 = a_5 + d = 5a + a = 6a \]
- Seventh term (\( a_7 \)):
\[ a_7 = a_6 + d = 6a + a = 7a \]

6. **Final Answer**: The next three terms are \( 5a, 6a, 7a \).

### Summary:
- The sequence \( a, 2a, 3a, 4a \) is an A.P. with a common difference of \( a \).
- The next three terms are \( 5a, 6a, 7a \).