Determine if the given sequences form arithmetic progressions and identify the common difference.

Let's solve each sequence one by one to determine whether it forms an arithmetic sequence.

An arithmetic sequence has a constant difference between consecutive terms, called the common difference (d).
To check:
- Compute the difference between each pair of consecutive terms.
- If all differences are equal → it’s arithmetic; otherwise, it’s not.

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1) 4, 0, -4, -8, -12, ...

Differences:
- 0 - 4 = -4
- -4 - 0 = -4
- -8 - (-4) = -4
- -12 - (-8) = -4

✔ All differences = -4 → Arithmetic sequence, d = -4

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2) 6.6, 7.2, 7.8, 8.4, 9, ...

Differences:
- 7.2 - 6.6 = 0.6
- 7.8 - 7.2 = 0.6
- 8.4 - 7.8 = 0.6
- 9 - 8.4 = 0.6

✔ All differences = 0.6 → Arithmetic sequence, d = 0.6

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3) 7, 10, 16, 19, 25, ...

Differences:
- 10 - 7 = 3
- 16 - 10 = 6
- 19 - 16 = 3
- 25 - 19 = 6

✘ Differences alternate: 3, 6, 3, 6 → Not constant → Not arithmetic

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4) 5√7, √7, -3√7, -7√7, -11√7, ...

Let’s factor out √7:
Sequence becomes:
5, 1, -3, -7, -11 → (all multiplied by √7)

Now compute differences:
- 1 - 5 = -4
- -3 - 1 = -4
- -7 - (-3) = -4
- -11 - (-7) = -4

✔ Constant difference = -4 → So original sequence is arithmetic with d = -4√7

→ Arithmetic sequence, d = -4√7

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5) -1.2, -1.4, -1.6, -1.8, -2, ...

Differences:
- -1.4 - (-1.2) = -0.2
- -1.6 - (-1.4) = -0.2
- -1.8 - (-1.6) = -0.2
- -2 - (-1.8) = -0.2

✔ All differences = -0.2 → Arithmetic sequence, d = -0.2

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6) 1, -6, 36, -216, 1296, ...

Differences:
- -6 - 1 = -7
- 36 - (-6) = 42
- -216 - 36 = -252
- 1296 - (-216) = 1512

✘ Differences: -7, 42, -252, 1512 → Not constant → Not arithmetic

(Note: This looks like a geometric sequence: multiply by -6 each time.)

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7) 1, 4, 8, 13, 19, ...

Differences:
- 4 - 1 = 3
- 8 - 4 = 4
- 13 - 8 = 5
- 19 - 13 = 6

✘ Differences: 3, 4, 5, 6 → Increasing → Not arithmetic

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8) 9, 14, 19, 24, 29, ...

Differences:
- 14 - 9 = 5
- 19 - 14 = 5
- 24 - 19 = 5
- 29 - 24 = 5

✔ All differences = 5 → Arithmetic sequence, d = 5

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9) -8, -15, -22, -29, -36, ...

Differences:
- -15 - (-8) = -7
- -22 - (-15) = -7
- -29 - (-22) = -7
- -36 - (-29) = -7

✔ All differences = -7 → Arithmetic sequence, d = -7

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10) 3, 9, 27, 81, 243, ...

Differences:
- 9 - 3 = 6
- 27 - 9 = 18
- 81 - 27 = 54
- 243 - 81 = 162

✘ Differences: 6, 18, 54, 162 → Multiplying by 3 → Not arithmetic

(This is geometric: multiply by 3 each time.)

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✔ Final Answers:

1) Yes, d = -4
2) Yes, d = 0.6
3) No
4) Yes, d = -4√7
5) Yes, d = -0.2
6) No
7) No
8) Yes, d = 5
9) Yes, d = -7
10) No

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Summary Table:

| Problem | Arithmetic? | Common Difference (d) |
|--------|-------------|-----------------------|
| 1 | Yes | -4 |
| 2 | Yes | 0.6 |
| 3 | No | – |
| 4 | Yes | -4√7 |
| 5 | Yes | -0.2 |
| 6 | No | – |
| 7 | No | – |
| 8 | Yes | 5 |
| 9 | Yes | -7 |
| 10 | No | – |

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