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

Answer

To determine if the sequence \( a, a^2, a^3, a^4, \ldots \) 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: \( a_2 = a^2 \)
- Third term: \( a_3 = a^3 \)
- Fourth term: \( a_4 = a^4 \)

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

3. **Simplify the Differences**:
- For \( d_1 \):
\[ d_1 = a^2 - a = a(a - 1) \]
- For \( d_2 \):
\[ d_2 = a^3 - a^2 = a^2(a - 1) \]
- For \( d_3 \):
\[ d_3 = a^4 - a^3 = a^3(a - 1) \]

4. **Check if the Differences are Equal**:
- We need to check if \( d_1 = d_2 \) and \( d_2 = d_3 \):
- \( d_1 = a(a - 1) \)
- \( d_2 = a^2(a - 1) \)
- \( d_3 = a^3(a - 1) \)

Since \( d_1 \), \( d_2 \), and \( d_3 \) are not equal for all values of \( a \) (except for specific cases like \( a = 1 \) or \( a = 0 \)), we conclude that the differences are not constant.

5. **Conclusion**: Since the differences are not equal, the sequence \( a, a^2, a^3, a^4, \ldots \) does not form an arithmetic progression (A.P.). Therefore, we cannot find a common difference or write three more terms.