https://file.aiquickdraw.com/mathbot/file/1728395792409j26rybyu.jpg

Let's go through each sequence one by one and determine the missing terms:

1. **Sequence**: 1, 4, 8, 16, 32, 64  
   - This is a geometric sequence where each term is multiplied by 2. The next term is \( 64 \times 2 = 128 \).
   
   **Answer**: 128

2. **Sequence**: 7, 21, 63, 189  
   - This is a geometric sequence where each term is multiplied by 3. The next term is \( 189 \times 3 = 567 \).
   
   **Answer**: 567

3. **Sequence**: 54, 18, 6, 2  
   - This is a geometric sequence where each term is divided by 3. The next term is \( 2 \div 3 = \frac{2}{3} \).
   
   **Answer**: \( \frac{2}{3} \)

4. **Sequence**: 10, 20, 40, 70  
   - The differences between terms are 10, 20, and 30 (i.e., an arithmetic sequence with a common difference that increases by 10). The next difference should be 40, so the next term is \( 70 + 40 = 110 \).
   
   **Answer**: 110

5. **Sequence**: 324, 108, 36, 12  
   - This is a geometric sequence where each term is divided by 3. The next term is \( 12 \div 3 = 4 \).
   
   **Answer**: 4

6. **Sequence**: a, b, a + b, a + 2b, 2a + 3b  
   - Observe the coefficients of \(a\) and \(b\) in each term:
     - \( a + 0b \), \( 0a + b \), \( a + b \), \( a + 2b \), \( 2a + 3b \).
     - Following the pattern, the next term should be \( 3a + 5b \).
   
   **Answer**: \( 3a + 5b \)

7. **Sequence**: 4, 4, 5, 6, 6, 7, 8, 8, 9  
   - The pattern seems to alternate between repeated numbers and increasing sequences. The next term should continue this pattern, so it would be 9 again.
   
   **Answer**: 9

8. **Sequence**: 2, 13, 8, 19, 14, 25, 20, 31, 26  
   - This pattern alternates between adding 11 and subtracting 5. After 26, add 11: \( 26 + 11 = 37 \).
   
   **Answer**: 37

9. **Sequence**: 22, 20, 17, 15, 12, 10, 7, 5, 2  
   - The differences are: -2, -3, -2, -3, etc. The next difference should be -3, so \( 2 - 3 = -1 \).
   
   **Answer**: -1

10. **Sequence**: 486, 162, 54, 18  
    - This is a geometric sequence where each term is divided by 3. The next term is \( 18 \div 3 = 6 \).
    
    **Answer**: 6

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Would you like further clarification or more details on any of these? Here are 5 related questions for deeper understanding:

1. How can we identify whether a sequence is arithmetic or geometric?
2. Can there be mixed sequences with both arithmetic and geometric properties?
3. What strategies can help in recognizing alternating patterns in sequences?
4. How do we deal with sequences involving variables, like in question 6?
5. What other mathematical patterns exist besides arithmetic and geometric sequences?

**Tip**: When working with sequences, always look for patterns in differences (arithmetic) or ratios (geometric) between terms.

Determine the missing term in each sequence from the given image.

Learn how to determine the missing terms in various arithmetic and geometric sequences with examples. This problem covers key concepts such as identifying patterns in sequences and applying arithmetic and geometric rules to find missing values. Suitable for students in Grades 6-8.

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