https://file.aiquickdraw.com/mathbot/file/1727085047965rup6spob.jpg

The problem shows two sequences, A and B, with mixed terms and asks to find the rule for each sequence based on certain conditions. Specifically:

- **Sequence A** has a step of 2 (i.e., it's an arithmetic sequence with a common difference of 2).
- **Sequence B** is another sequence, but the step for it is not explicitly provided.

The jumbled terms are:  
\[ 14, 9, 22, 13, 20, 16, 21, 18, 5, 17 \]

### Step 1: Identify Sequence A
Since Sequence A increases by 2 each time, we should find a sequence from the given numbers that follows this rule.

- Starting with **9** (smallest number), the next term would be \(9 + 2 = 11\), but **11** is not available. Let's instead try **9** and jump to the next possible terms.
- \(9 + 4 = 13\), so **13** works.
- \(13 + 2 = 15\) (but 15 is unavailable), jump to **16** next.
- \(16 + 2 = 18\), and **18** works.
- \(18 + 2 = 20\), and **20** works.

Thus, **Sequence A** is:  
\[ 9, 13, 16, 18, 20 \]

### Step 2: Identify Sequence B
The remaining terms form Sequence B, which increases by a different (unknown) step. The remaining numbers are:
\[ 5, 14, 17, 21, 22 \]

- Start with **5** (smallest number).
- Then, **14** is the next in the sequence.
- Continue to **17**, **21**, and finally **22**.

Thus, **Sequence B** is:  
\[ 5, 14, 17, 21, 22 \]

### Sequence Rules:
- **Sequence A**: Starts at **9** and increases by **2** each time.
- **Sequence B**: Starts at **5**, but the common difference isn't constant; hence it may follow a different pattern.

Would you like a further breakdown, or do you have any questions on this?

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### 5 Related Questions:
1. What is an arithmetic sequence, and how can you recognize one?
2. How can we generalize the formula for an arithmetic sequence?
3. What are the typical steps for identifying a non-arithmetic sequence like Sequence B?
4. How can we graph the terms of Sequence A and Sequence B to visually compare them?
5. What if the step size of Sequence B were explicitly given — how would this change the method?

### Tip:
When solving for sequences, always check for patterns (e.g., constant differences or other trends) before assuming the type of sequence you're dealing with.

The first five terms from two increasing sequences have been jumbled up. Sequence A has equal steps of 2. By separating out the terms, find the rule for each sequence.

Learn how to separate and identify two increasing arithmetic sequences from jumbled terms. Sequence A follows a consistent step of 2, while Sequence B has a different pattern. Suitable for students in Grades 6-8.

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