Analyze the given sequence for its rule and identify the next three terms.

a. 1, 10, 100, 1000

b. 2, 5, 9, 14, 20

Let's analyze the two sequences one by one:

### Sequence a: 1, 10, 100, 1000
- **Rule**: Each term in the sequence is obtained by multiplying the previous term by 10.
  - \(1 \times 10 = 10\)
  - \(10 \times 10 = 100\)
  - \(100 \times 10 = 1000\)
  
- **Next three terms**:
  - \(1000 \times 10 = 10000\)
  - \(10000 \times 10 = 100000\)
  - \(100000 \times 10 = 1000000\)

Thus, the next three terms are **10,000, 100,000, and 1,000,000**.

### Sequence b: 2, 5, 9, 14, 20
- **Rule**: To find the difference between consecutive terms:
  - \(5 - 2 = 3\)
  - \(9 - 5 = 4\)
  - \(14 - 9 = 5\)
  - \(20 - 14 = 6\)
  
  The differences between consecutive terms are increasing by 1. This suggests the sequence is based on adding an increasing value (starting from 3) to each term to get the next one.

- **Next three terms**:
  - \(20 + 7 = 27\)  (add 7 to the last term)
  - \(27 + 8 = 35\)  (add 8)
  - \(35 + 9 = 44\)  (add 9)

Thus, the next three terms are **27, 35, and 44**.

Would you like more details or have any questions?

### Related Questions:
1. How do geometric sequences differ from arithmetic sequences?
2. Can the first sequence be represented using an algebraic formula?
3. What is the general term (nth term) for the second sequence?
4. How can you identify the type of sequence from a given list of numbers?
5. What are common applications of geometric sequences in real life?

**Tip**: When analyzing sequences, always check the differences or ratios between consecutive terms to identify the rule governing the sequence.

Understand the rules governing arithmetic and geometric sequences. Analyze sequences like 1, 10, 100, 1000 and 2, 5, 9, 14, 20 to identify their next three terms. Detailed explanations provided for each sequence.

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