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),5^(2),7^(2),73,…..`

Answer

To determine whether the given sequence \(1^2, 5^2, 7^2, 73, \ldots\) forms an arithmetic progression (A.P.), we will follow these steps:

### Step 1: Calculate the terms
First, we need to evaluate the squares of the numbers:
- \(1^2 = 1\)
- \(5^2 = 25\)
- \(7^2 = 49\)

So, the sequence becomes:
\[1, 25, 49, 73\]

### Step 2: Find the differences between consecutive terms
Next, we will find the differences between consecutive terms to check if they are constant:
- Difference between the first and second term:
\[ 25 - 1 = 24 \]
- Difference between the second and third term:
\[ 49 - 25 = 24 \]
- Difference between the third and fourth term:
\[ 73 - 49 = 24 \]

### Step 3: Check if the common difference is constant
Since the differences are:
- \(24\)
- \(24\)
- \(24\)

The common difference is constant and equal to \(24\). Therefore, the sequence is an arithmetic progression (A.P.).

### Step 4: Write three more terms
To find the next three terms in the A.P., we will keep adding the common difference \(24\) to the last term \(73\):
- First additional term:
\[ 73 + 24 = 97 \]
- Second additional term:
\[ 97 + 24 = 121 \]
- Third additional term:
\[ 121 + 24 = 145 \]

### Final Result
The sequence is an A.P. with a common difference of \(24\). The next three terms are:
\[97, 121, 145\]