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, -3.2, - 5.2, -7.2,….`

Answer

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

### Step 1: Identify the given terms
The given sequence is:
-1.2, -3.2, -5.2, -7.2

### Step 2: Calculate the common difference (d)
To check if the sequence is an AP, we need to find the common difference (d) between consecutive terms. The common difference is calculated as follows:

1. **Calculate d1 = a2 - a1**
- a1 = -1.2
- a2 = -3.2
- d1 = -3.2 - (-1.2) = -3.2 + 1.2 = -2.0

2. **Calculate d2 = a3 - a2**
- a2 = -3.2
- a3 = -5.2
- d2 = -5.2 - (-3.2) = -5.2 + 3.2 = -2.0

3. **Calculate d3 = a4 - a3**
- a3 = -5.2
- a4 = -7.2
- d3 = -7.2 - (-5.2) = -7.2 + 5.2 = -2.0

Since d1, d2, and d3 are all equal to -2.0, we can conclude that the sequence is an AP.

### Step 3: State the common difference
The common difference (d) is:
d = -2.0

### Step 4: Find the next three terms
To find the next three terms (a5, a6, a7), we can use the formula for the nth term of an AP:
\[ a_n = a_1 + (n-1)d \]

1. **Calculate a5**
- a5 = a1 + 4d
- a5 = -1.2 + 4(-2.0) = -1.2 - 8 = -9.2

2. **Calculate a6**
- a6 = a1 + 5d
- a6 = -1.2 + 5(-2.0) = -1.2 - 10 = -11.2

3. **Calculate a7**
- a7 = a1 + 6d
- a7 = -1.2 + 6(-2.0) = -1.2 - 12 = -13.2

### Step 5: Write the next three terms
The next three terms are:
- a5 = -9.2
- a6 = -11.2
- a7 = -13.2

### Final Answer
The sequence is an AP with a common difference of -2.0. The next three terms are -9.2, -11.2, and -13.2.

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