Which of the following are AP's ? If they form an A. P. , find the common difference d and write three more terms.
`-10,-6,-2,2…`

Answer

To determine if the sequence \(-10, -6, -2, 2\) forms an Arithmetic Progression (AP), we will follow these steps:

### Step 1: Identify the terms
Let:
- \( A_1 = -10 \)
- \( A_2 = -6 \)
- \( A_3 = -2 \)
- \( A_4 = 2 \)

### Step 2: Calculate the common difference \(d\)
The common difference \(d\) in an AP is calculated as the difference between consecutive terms.

1. Calculate \(d\) between \(A_1\) and \(A_2\):
\[ d = A_2 - A_1 = -6 - (-10) = -6 + 10 = 4 \]

2. Calculate \(d\) between \(A_2\) and \(A_3\):
\[ d = A_3 - A_2 = -2 - (-6) = -2 + 6 = 4 \]

3. Calculate \(d\) between \(A_3\) and \(A_4\):
\[ d = A_4 - A_3 = 2 - (-2) = 2 + 2 = 4 \]

Since the common difference \(d\) is the same (4) for all pairs of consecutive terms, we can conclude that the sequence is indeed an AP.

### Step 3: Write 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:
\[ A_n = A_1 + (n-1)d \]

1. Calculate \(A_5\):
\[ A_5 = A_1 + 4d = -10 + 4 \times 4 = -10 + 16 = 6 \]

2. Calculate \(A_6\):
\[ A_6 = A_1 + 5d = -10 + 5 \times 4 = -10 + 20 = 10 \]

3. Calculate \(A_7\):
\[ A_7 = A_1 + 6d = -10 + 6 \times 4 = -10 + 24 = 14 \]

### Conclusion
The next three terms in the sequence are \(6\), \(10\), and \(14\).

### Summary
- The given sequence \(-10, -6, -2, 2\) is an AP.
- The common difference \(d\) is \(4\).
- The next three terms are \(6\), \(10\), and \(14\).

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