https://file.aiquickdraw.com/mathbot/file/1726629074157fqiuxt3p.jpg

Let's break down the problem and complete the tasks one by one.

### a. Completing the table for each sequence:

From the graph, we can extract the values of the terms for both Sequence A (black points) and Sequence B (blue points).

- **Sequence A (Black dots)**:
  - At term 0: \(-1\)
  - At term 1: \(0\)
  - At term 2: \(3\)
  - At term 3: \(8\)
  - At term 4: \(15\)
  - At term 5: \(24\)
  - At term 6: \(35\)

- **Sequence B (Blue dots)**:
  - At term 0: \(\frac{1}{2}\)
  - At term 1: \(2\)
  - At term 2: \(8\)
  - At term 3: \(18\)
  - At term 4: \(32\)
  - At term 5: \(50\)
  - At term 6: \(72\)

Now, the table can be completed as follows:

| Term Number | Sequence A | Sequence B |
|-------------|------------|------------|
| 0           | -1         | \(\frac{1}{2}\) |
| 1           | 0          | 2          |
| 2           | 3          | 8          |
| 3           | 8          | 18         |
| 4           | 15         | 32         |
| 5           | 24         | 50         |
| 6           | 35         | 72         |

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### b. For Sequence A: Describe a way to produce a new term from the previous term.

For **Sequence A**, the difference between consecutive terms increases by 2 each time:

- \(0 - (-1) = 1\)
- \(3 - 0 = 3\)
- \(8 - 3 = 5\)
- \(15 - 8 = 7\)
- \(24 - 15 = 9\)
- \(35 - 24 = 11\)

This pattern suggests that to generate the next term, you add an increasing odd number to the previous term (starting from 1 and increasing by 2 each time). 

### c. For Sequence B: Describe a way to produce a new term from the previous term.

For **Sequence B**, the difference between consecutive terms also increases:

- \(2 - \frac{1}{2} = \frac{3}{2}\)
- \(8 - 2 = 6\)
- \(18 - 8 = 10\)
- \(32 - 18 = 14\)
- \(50 - 32 = 18\)
- \(72 - 50 = 22\)

The difference between terms is increasing by 4 each time, suggesting the following rule: to produce a new term in Sequence B, add a number to the previous term, where that number increases by 4 each time.

### d. Which of these is a geometric sequence? Explain how you know.

A **geometric sequence** is one in which each term is produced by multiplying the previous term by a constant ratio. Neither of these sequences appears to be geometric because:
- Sequence A shows an increasing difference, which suggests it is more like an arithmetic sequence with an increasing increment.
- Sequence B also shows increasing differences rather than a constant ratio, ruling out it being geometric.

Thus, **neither sequence is geometric**.

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Let me know if you'd like more details on any of these solutions or any further clarification. Here are five related questions for further exploration:

1. What type of sequence is Sequence A? How can you define it algebraically?
2. Can you express Sequence B as a quadratic function?
3. What would be the 7th term in Sequence A if the pattern continues?
4. If Sequence B continues, what would be the value of the 7th term?
5. How would you generalize the pattern of differences for Sequence B?

**Tip**: Always check the difference or ratio between consecutive terms when trying to identify the type of sequence (arithmetic, geometric, etc.).

Here is the graph of two sequences. Complete the table for each sequence. Describe how to produce new terms for Sequence A and B from the previous term. Which of these is a geometric sequence?

Learn how to complete the table for two sequences and identify patterns for generating new terms. Understand the difference between arithmetic and geometric sequences with real examples and step-by-step explanations. Suitable for students in Grades 8-10.

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