Which of the following are AP's ? If they form an A. P. , find the common difference d and write three more terms.
`-0,-4,-8,-12,…`

Answer

To determine whether the sequence \(-0, -4, -8, -12, \ldots\) forms an arithmetic progression (AP), we will follow these steps:

### Step 1: Identify the terms of the sequence
The terms given are:
- \(a_1 = -0\)
- \(a_2 = -4\)
- \(a_3 = -8\)
- \(a_4 = -12\)

### Step 2: Calculate the common difference
The common difference \(d\) in an AP is calculated by subtracting any term from the subsequent term. We can calculate \(d\) using the first two terms:

\[ d = a_2 - a_1 = -4 - (-0) = -4 \]

Now, let's verify this with the next pair of terms:

\[ d = a_3 - a_2 = -8 - (-4) = -8 + 4 = -4 \]

And again with the next pair:

\[ d = a_4 - a_3 = -12 - (-8) = -12 + 8 = -4 \]

Since the common difference \(d\) is the same for all pairs, we confirm that the sequence is an AP.

### Step 3: Write the common difference
The common difference \(d\) is:

\[ d = -4 \]

### Step 4: Find the next three terms
To find the next three terms \(a_5\), \(a_6\), and \(a_7\), we use 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.

- For \(a_5\):
\[ a_5 = a_1 + 4d = -0 + 4(-4) = -0 - 16 = -16 \]

- For \(a_6\):
\[ a_6 = a_1 + 5d = -0 + 5(-4) = -0 - 20 = -20 \]

- For \(a_7\):
\[ a_7 = a_1 + 6d = -0 + 6(-4) = -0 - 24 = -24 \]

### Conclusion
The next three terms of the AP are:
- \(a_5 = -16\)
- \(a_6 = -20\)
- \(a_7 = -24\)

### Final Answer
The sequence \(-0, -4, -8, -12, \ldots\) is an AP with a common difference of \(-4\), and the next three terms are \(-16, -20, -24\).

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