Arithmetic sequence worksheet for identifying common differences in given number sequences.

To determine whether the given sequences form arithmetic progressions (APs), we need to check if the difference between consecutive terms is constant. If the difference is constant, the sequence is an arithmetic progression, and that constant difference is called the common difference \( d \).

Let's analyze each sequence step by step:

---

1) \( 4, 0, -4, -8, -12, \ldots \)

- Calculate the differences between consecutive terms:
\[
0 - 4 = -4
\]
\[
-4 - 0 = -4
\]
\[
-8 - (-4) = -4
\]
\[
-12 - (-8) = -4
\]

- The differences are all equal to \( -4 \).

- Conclusion: This is an arithmetic progression with a common difference \( d = -4 \).

---

2) \( 6.6, 7.2, 7.8, 8.4, 9, \ldots \)

- Calculate the differences between consecutive terms:
\[
7.2 - 6.6 = 0.6
\]
\[
7.8 - 7.2 = 0.6
\]
\[
8.4 - 7.8 = 0.6
\]
\[
9 - 8.4 = 0.6
\]

- The differences are all equal to \( 0.6 \).

- Conclusion: This is an arithmetic progression with a common difference \( d = 0.6 \).

---

3) \( 7, 10, 16, 19, 25, \ldots \)

- Calculate the differences between consecutive terms:
\[
10 - 7 = 3
\]
\[
16 - 10 = 6
\]
\[
19 - 16 = 3
\]
\[
25 - 19 = 6
\]

- The differences are not constant (\( 3, 6, 3, 6, \ldots \)).

- Conclusion: This is not an arithmetic progression.

---

4) \( 5\sqrt{2}, \sqrt{2}, -3\sqrt{2}, -7\sqrt{2}, -11\sqrt{2}, \ldots \)

- Calculate the differences between consecutive terms:
\[
\sqrt{2} - 5\sqrt{2} = -4\sqrt{2}
\]
\[
-3\sqrt{2} - \sqrt{2} = -4\sqrt{2}
\]
\[
-7\sqrt{2} - (-3\sqrt{2}) = -4\sqrt{2}
\]
\[
-11\sqrt{2} - (-7\sqrt{2}) = -4\sqrt{2}
\]

- The differences are all equal to \( -4\sqrt{2} \).

- Conclusion: This is an arithmetic progression with a common difference \( d = -4\sqrt{2} \).

---

5) \( -1.2, -1.4, -1.6, -1.8, -2, \ldots \)

- Calculate the differences between consecutive terms:
\[
-1.4 - (-1.2) = -0.2
\]
\[
-1.6 - (-1.4) = -0.2
\]
\[
-1.8 - (-1.6) = -0.2
\]
\[
-2 - (-1.8) = -0.2
\]

- The differences are all equal to \( -0.2 \).

- Conclusion: This is an arithmetic progression with a common difference \( d = -0.2 \).

---

6) \( 1, -6, 36, -216, 1296, \ldots \)

- Calculate the ratios between consecutive terms:
\[
\frac{-6}{1} = -6
\]
\[
\frac{36}{-6} = -6
\]
\[
\frac{-216}{36} = -6
\]
\[
\frac{1296}{-216} = -6
\]

- The ratios are constant, but the differences are not:
\[
-6 - 1 = -7
\]
\[
36 - (-6) = 42
\]
\[
-216 - 36 = -252
\]
\[
1296 - (-216) = 1512
\]

- Conclusion: This is not an arithmetic progression (it is a geometric progression).

---

7) \( 1, 4, 8, 13, 19, \ldots \)

- Calculate the differences between consecutive terms:
\[
4 - 1 = 3
\]
\[
8 - 4 = 4
\]
\[
13 - 8 = 5
\]
\[
19 - 13 = 6
\]

- The differences are not constant (\( 3, 4, 5, 6, \ldots \)).

- Conclusion: This is not an arithmetic progression.

---

8) \( 9, 14, 19, 24, 29, \ldots \)

- Calculate the differences between consecutive terms:
\[
14 - 9 = 5
\]
\[
19 - 14 = 5
\]
\[
24 - 19 = 5
\]
\[
29 - 24 = 5
\]

- The differences are all equal to \( 5 \).

- Conclusion: This is an arithmetic progression with a common difference \( d = 5 \).

---

9) \( -8, -15, -22, -29, -36, \ldots \)

- Calculate the differences between consecutive terms:
\[
-15 - (-8) = -7
\]
\[
-22 - (-15) = -7
\]
\[
-29 - (-22) = -7
\]
\[
-36 - (-29) = -7
\]

- The differences are all equal to \( -7 \).

- Conclusion: This is an arithmetic progression with a common difference \( d = -7 \).

---

10) \( 3, 9, 27, 81, 243, \ldots \)

- Calculate the ratios between consecutive terms:
\[
\frac{9}{3} = 3
\]
\[
\frac{27}{9} = 3
\]
\[
\frac{81}{27} = 3
\]
\[
\frac{243}{81} = 3
\]

- The ratios are constant, but the differences are not:
\[
9 - 3 = 6
\]
\[
27 - 9 = 18
\]
\[
81 - 27 = 54
\]
\[
243 - 81 = 162
\]

- Conclusion: This is not an arithmetic progression (it is a geometric progression).

---

Final Answers

1. Arithmetic progression, \( d = -4 \)
2. Arithmetic progression, \( d = 0.6 \)
3. Not an arithmetic progression
4. Arithmetic progression, \( d = -4\sqrt{2} \)
5. Arithmetic progression, \( d = -0.2 \)
6. Not an arithmetic progression
7. Not an arithmetic progression
8. Arithmetic progression, \( d = 5 \)
9. Arithmetic progression, \( d = -7 \)
10. Not an arithmetic progression

\[
\boxed{
\begin{array}{ll}
1) & \text{Arithmetic progression, } d = -4 \\
2) & \text{Arithmetic progression, } d = 0.6 \\
3) & \text{Not an arithmetic progression} \\
4) & \text{Arithmetic progression, } d = -4\sqrt{2} \\
5) & \text{Arithmetic progression, } d = -0.2 \\
6) & \text{Not an arithmetic progression} \\
7) & \text{Not an arithmetic progression} \\
8) & \text{Arithmetic progression, } d = 5 \\
9) & \text{Arithmetic progression, } d = -7 \\
10) & \text{Not an arithmetic progression}
\end{array}
}
\]