https://file.aiquickdraw.com/mathbot/file/1726703345103movge0p4.jpg

Let's analyze both sequences in the image to determine if they are **arithmetic** or **geometric**.

### Sequence a: \(1, -2, 4, -8, \dots \)

1. **Check for Arithmetic Sequence**:
   In an arithmetic sequence, the difference between consecutive terms is constant.

   - \( -2 - 1 = -3 \)
   - \( 4 - (-2) = 6 \)
   - \( -8 - 4 = -12 \)

   Since the differences are not the same (\(-3, 6, -12\)), it is **not an arithmetic sequence**.

2. **Check for Geometric Sequence**:
   In a geometric sequence, the ratio between consecutive terms is constant.

   - \( \frac{-2}{1} = -2 \)
   - \( \frac{4}{-2} = -2 \)
   - \( \frac{-8}{4} = -2 \)

   The ratio is consistently \(-2\), so this is a **geometric sequence** with a common ratio of \(-2\).

---

### Sequence b: \(1, -2, -5, -8, \dots\)

1. **Check for Arithmetic Sequence**:
   - \( -2 - 1 = -3 \)
   - \( -5 - (-2) = -3 \)
   - \( -8 - (-5) = -3 \)

   The difference between consecutive terms is consistently \(-3\), so this is an **arithmetic sequence** with a common difference of \(-3\).

2. **Check for Geometric Sequence**:
   We don't need to check for a geometric sequence since we've already confirmed it's arithmetic, but for completeness:
   - \( \frac{-2}{1} = -2 \)
   - \( \frac{-5}{-2} = 2.5 \)
   - \( \frac{-8}{-5} = 1.6 \)

   The ratios are inconsistent, confirming it's not geometric.

---

### Conclusion:
- Sequence (a) is **geometric** with a common ratio of \(-2\).
- Sequence (b) is **arithmetic** with a common difference of \(-3\).

Would you like further details or have any questions about these sequences?

Here are five related questions:
1. How do you identify the common ratio in a geometric sequence?
2. What is the general formula for the \(n\)-th term of an arithmetic sequence?
3. Can a sequence be both arithmetic and geometric? Why or why not?
4. How do you derive the sum of a finite geometric series?
5. What are some applications of arithmetic and geometric sequences in real life?

**Tip:** In a geometric sequence, negative common ratios result in alternating positive and negative terms!

Decide whether each sequence is arithmetic or geometric: a) 1, -2, 4, -8,... b) 1, -2, -5, -8,...

Learn how to decide whether sequences are arithmetic or geometric. For example, sequence a) 1, -2, 4, -8,... is geometric with a common ratio of -2, and sequence b) 1, -2, -5, -8,... is arithmetic with a common difference of -3. Suitable for students in Grades 6-8.

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