Which of the following are AP's ? If they form an A. P. , find the common difference d and write three more terms.
`1 ^(2) , 5 ^(2), 7 ^(2) , 73,…`

Answer

To determine if the given sequence \(1^2, 5^2, 7^2, 73, \ldots\) forms an Arithmetic Progression (AP), we will follow these steps:

### Step 1: Calculate the terms of the sequence
First, we need to calculate the squares of the numbers given in the sequence:
- \(1^2 = 1\)
- \(5^2 = 25\)
- \(7^2 = 49\)
- The fourth term is given as \(73\).

So, the sequence we have is:
\[1, 25, 49, 73\]

### Step 2: Find the common difference \(d\)
To check if the sequence is an AP, we need to find the difference between consecutive terms.

1. Calculate \(a_2 - a_1\):
\[ d_1 = 25 - 1 = 24 \]

2. Calculate \(a_3 - a_2\):
\[ d_2 = 49 - 25 = 24 \]

3. Calculate \(a_4 - a_3\):
\[ d_3 = 73 - 49 = 24 \]

Since the differences \(d_1\), \(d_2\), and \(d_3\) are all equal to \(24\), we conclude that the sequence is indeed an AP with a common difference \(d = 24\).

### Step 3: Write the next three terms
Now that we know the sequence is an AP, we can find the next three terms \(a_5\), \(a_6\), and \(a_7\) using the formula for the \(n\)-th term of an AP:
\[ a_n = a + (n-1)d \]
where \(a\) is the first term and \(d\) is the common difference.

1. Calculate \(a_5\):
\[ a_5 = a + 4d = 1 + 4 \times 24 = 1 + 96 = 97 \]

2. Calculate \(a_6\):
\[ a_6 = a + 5d = 1 + 5 \times 24 = 1 + 120 = 121 \]

3. Calculate \(a_7\):
\[ a_7 = a + 6d = 1 + 6 \times 24 = 1 + 144 = 145 \]

### Final Answer
The next three terms of the sequence are:
\[97, 121, 145\]