To determine which sequences are arithmetic sequences, we need to check if there is a constant difference between consecutive terms in each sequence. An arithmetic sequence has a common difference $d$ such that each term is obtained by adding $d$ to the previous term.

Let’s analyze each option:

Option a: $3, 1, -1, -3, \dots$

• Difference between $3$ and $1$: $1 - 3 = -2$
• Difference between $1$ and $-1$: $-1 - 1 = -2$
• Difference between $-1$ and $-3$: $-3 - (-1) = -2$

Conclusion: This is an arithmetic sequence with common difference $d = -2$.

Option b: $15, 11, 7, 3, \dots$

• Difference between $15$ and $11$: $11 - 15 = -4$
• Difference between $11$ and $7$: $7 - 11 = -4$
• Difference between $7$ and $3$: $3 - 7 = -4$

Conclusion: This is an arithmetic sequence with common difference $d = -4$.

Option c: $20, 10, 5, \frac{5}{2}, \dots$

• Difference between $20$ and $10$: $10 - 20 = -10$
• Difference between $10$ and $5$: $5 - 10 = -5$

The differences are not consistent.

Conclusion: This is not an arithmetic sequence.

Option d: $1, -3, 9, -27, \dots$

• Difference between $1$ and $-3$: $-3 - 1 = -4$
• Difference between $-3$ and $9$: $9 - (-3) = 12$

The differences are not consistent.

Conclusion: This is not an arithmetic sequence (it's actually a geometric sequence).

Option e: $3 \log 4, 3 \log 8, 3 \log 16, \dots$

Rewrite each term using properties of logarithms:

• $3 \log 4 = 3 \times 2 = 6$
• $3 \log 8 = 3 \times 3 = 9$
• $3 \log 16 = 3 \times 4 = 12$

Each term has a difference of $+3$.

Conclusion: This is an arithmetic sequence with common difference $d = 3$.

Option f: $2 + \sqrt{2}, 2 + 2\sqrt{2}, 2 + 3\sqrt{2}, \dots$

• Difference between $2 + \sqrt{2}$ and $2 + 2\sqrt{2}$: $(2 + 2\sqrt{2}) - (2 + \sqrt{2}) = \sqrt{2}$
• Difference between $2 + 2\sqrt{2}$ and $2 + 3\sqrt{2}$: $(2 + 3\sqrt{2}) - (2 + 2\sqrt{2}) = \sqrt{2}$

Each term has a difference of $\sqrt{2}$.

Conclusion: This is an arithmetic sequence with common difference $d = \sqrt{2}$.

Summary

The arithmetic sequences among the options are:

• a: $3, 1, -1, -3, \dots$ with $d = -2$
• b: $15, 11, 7, 3, \dots$ with $d = -4$
• e: $3 \log 4, 3 \log 8, 3 \log 16, \dots$ with $d = 3$
• f: $2 + \sqrt{2}, 2 + 2\sqrt{2}, 2 + 3\sqrt{2}, \dots$ with $d = \sqrt{2}$

Would you like further details on any part of this solution, or do you have any questions?

Here are 5 additional questions related to arithmetic sequences for further exploration:

1. How do you find the general formula for an arithmetic sequence?
2. What is the sum of the first $n$ terms of an arithmetic sequence?
3. How can you determine if a sequence is geometric rather than arithmetic?
4. What are some real-life applications of arithmetic sequences?
5. How does an arithmetic sequence differ from an arithmetic progression?

Tip: Always check the consistency of the differences between terms to verify if a sequence is arithmetic.