Which of the following form an ap ? Justify your answer
0, 0, 1, 1, 2, 2, 3, 3 ............

Answer

To determine whether the sequence \( 0, 0, 1, 1, 2, 2, 3, 3, \ldots \) forms an arithmetic progression (AP), we will follow these steps:

### Step 1: Define the terms of the sequence
Let’s denote the terms of the sequence as follows:
- \( A_1 = 0 \)
- \( A_2 = 0 \)
- \( A_3 = 1 \)
- \( A_4 = 1 \)
- \( A_5 = 2 \)
- \( A_6 = 2 \)
- \( A_7 = 3 \)
- \( A_8 = 3 \)

### Step 2: Calculate the differences between consecutive terms
We need to find the common difference \( d \) by calculating the differences between consecutive terms:
- \( d_1 = A_2 - A_1 = 0 - 0 = 0 \)
- \( d_2 = A_3 - A_2 = 1 - 0 = 1 \)
- \( d_3 = A_4 - A_3 = 1 - 1 = 0 \)
- \( d_4 = A_5 - A_4 = 2 - 1 = 1 \)
- \( d_5 = A_6 - A_5 = 2 - 2 = 0 \)
- \( d_6 = A_7 - A_6 = 3 - 2 = 1 \)
- \( d_7 = A_8 - A_7 = 3 - 3 = 0 \)

### Step 3: Compare the differences
Now, we will compare the calculated differences:
- The differences are \( 0, 1, 0, 1, 0, 1, 0 \).

### Step 4: Determine if the differences are constant
For a sequence to be an arithmetic progression, the common difference must be the same for all consecutive terms. In this case, the differences alternate between \( 0 \) and \( 1 \).

### Conclusion
Since the differences are not constant (they alternate between \( 0 \) and \( 1 \)), the sequence \( 0, 0, 1, 1, 2, 2, 3, 3, \ldots \) does not form an arithmetic progression.