Write of the following are A.P.s ? If they form an A.P., find the common difference and write three more terms .
`1^(2),3^(2),5^(2),7^(2)`

Answer

To determine if the sequence \(1^2, 3^2, 5^2, 7^2\) forms an arithmetic progression (A.P.), we will follow these steps:

### Step 1: Calculate the terms of the sequence
First, we will calculate the squares of the numbers:
- \(1^2 = 1\)
- \(3^2 = 9\)
- \(5^2 = 25\)
- \(7^2 = 49\)

Thus, the sequence is \(1, 9, 25, 49\).

### Step 2: Find the common difference
Next, we will find the common difference between consecutive terms:
- The difference between the first and second terms:
\[ 9 - 1 = 8 \]
- The difference between the second and third terms:
\[ 25 - 9 = 16 \]
- The difference between the third and fourth terms:
\[ 49 - 25 = 24 \]

### Step 3: Check if the common difference is constant
Now, we will check if the common differences are equal:
- The first difference is \(8\).
- The second difference is \(16\).
- The third difference is \(24\).

Since the common differences \(8\), \(16\), and \(24\) are not equal, we conclude that the sequence does not form an A.P.

### Conclusion
Since the common difference is not constant, we can conclude that the sequence \(1^2, 3^2, 5^2, 7^2\) does not form an arithmetic progression. Therefore, we cannot find a common difference or write three more terms.

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