https://file.aiquickdraw.com/mathbot/file/17258982498763kzqvfpq.jpg

Let's analyze and solve the patterns presented in the image.

1. **Pattern a: 3, 6, 9, ___ , ___**
   - The difference between terms is +3.
   - The next two terms will be: \( 9 + 3 = 12 \) and \( 12 + 3 = 15 \).
   - So, the sequence becomes: **3, 6, 9, 12, 15**.

2. **Pattern b: 3, 6, 12, ___ , ___**
   - The first difference is +3, and then +6.
   - This seems to follow a doubling pattern for each step.
   - The next difference will be \( +12 \) and \( +24 \).
   - So, \( 12 + 12 = 24 \) and \( 24 + 24 = 48 \).
   - The sequence becomes: **3, 6, 12, 24, 48**.

3. **Pattern c: 1, 9, 13, ___ , ___**
   - The difference between the first two terms is \( +8 \), and between the second and third is \( +4 \).
   - The difference appears to be decreasing by 4 each time.
   - The next difference would be \( +2 \), so \( 13 + 2 = 15 \).
   - Then, the difference would be \( +1 \), so \( 15 + 1 = 16 \).
   - The sequence becomes: **1, 9, 13, 15, 16**.

4. **Pattern d: 128, 64, ___ , ___ , 8**
   - The pattern appears to be dividing by 2.
   - \( 128 \div 2 = 64 \), and \( 64 \div 2 = 32 \).
   - Continuing this pattern: \( 32 \div 2 = 16 \) and \( 16 \div 2 = 8 \).
   - The sequence becomes: **128, 64, 32, 16, 8**.

### Final Answer:
- a. **12, 15**
- b. **24, 48**
- c. **15, 16**
- d. **32, 16**

Would you like more details on any of these patterns?

Here are 5 related questions to think about:
1. What would the 6th term in sequence b be?
2. Can you create your own pattern using subtraction?
3. How would the sequences change if we added a constant instead of increasing it each time?
4. What happens to pattern d if we change the division to multiplication?
5. Can you identify any other number sequences that might fit these patterns?

**Tip:** Identifying differences between terms is often the key to finding the rule behind a sequence!

Fill in the blanks to continue the patterns: a) 3, 6, 9, ___, ___ b) 3, 6, 12, ___, ___ c) 1, 9, 13, ___, ___ d) 128, 64, ___, ___, 8

Learn how to solve number pattern problems using arithmetic and geometric sequences. This example covers sequences such as 3, 6, 9 and 128, 64, using arithmetic progression and pattern recognition. Suitable for students in Grades 4-6.

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