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), - (1)/(2), - (1)/(2) , - (1)/(2),….`

Answer

To determine if the given sequence forms an Arithmetic Progression (AP) and to find the common difference and three additional terms, we will follow these steps:

### Step 1: Identify the terms of the sequence
The given sequence is:
- \( A_1 = -\frac{1}{2} \)
- \( A_2 = -\frac{1}{2} \)
- \( A_3 = -\frac{1}{2} \)
- \( A_4 = -\frac{1}{2} \)

### Step 2: Calculate the common difference \( d \)
The common difference \( d \) in an AP is calculated as:
\[ d = A_2 - A_1 \]
Substituting the values:
\[ d = -\frac{1}{2} - \left(-\frac{1}{2}\right) = -\frac{1}{2} + \frac{1}{2} = 0 \]

### Step 3: Verify the common difference with other terms
We can check the common difference between other consecutive terms:
\[ d = A_3 - A_2 = -\frac{1}{2} - \left(-\frac{1}{2}\right) = 0 \]
\[ d = A_4 - A_3 = -\frac{1}{2} - \left(-\frac{1}{2}\right) = 0 \]

Since the common difference \( d \) is the same (0) for all pairs of consecutive terms, we confirm that the sequence is indeed an AP.

### Step 4: Write three more terms
To find the next three terms of the AP, we can 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.

1. For the 5th term \( A_5 \):
\[ A_5 = A_1 + 4d = -\frac{1}{2} + 4 \cdot 0 = -\frac{1}{2} \]

2. For the 6th term \( A_6 \):
\[ A_6 = A_1 + 5d = -\frac{1}{2} + 5 \cdot 0 = -\frac{1}{2} \]

3. For the 7th term \( A_7 \):
\[ A_7 = A_1 + 6d = -\frac{1}{2} + 6 \cdot 0 = -\frac{1}{2} \]

### Conclusion
The next three terms are:
- \( A_5 = -\frac{1}{2} \)
- \( A_6 = -\frac{1}{2} \)
- \( A_7 = -\frac{1}{2} \)

Thus, the sequence is an AP with a common difference \( d = 0 \), and all terms are \( -\frac{1}{2} \).

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