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### A: \( 45, 90, 180, 360, \underline{\ \ }, \underline{\ \ }, \dots \)
- **Describe Pattern:** The numbers are doubling each time.
  - \( 45 \times 2 = 90 \)
  - \( 90 \times 2 = 180 \)
  - \( 180 \times 2 = 360 \)
- The next terms would be:
  - \( 360 \times 2 = 720 \)
  - \( 720 \times 2 = 1440 \)
- **Type of Sequence:** Geometric (ratio = 2)

### B: \( -4, -2, 0, 2, \underline{\ \ }, \underline{\ \ }, \dots \)
- **Describe Pattern:** The numbers are increasing by 2 each time.
  - \( -4 + 2 = -2 \)
  - \( -2 + 2 = 0 \)
  - \( 0 + 2 = 2 \)
- The next terms would be:
  - \( 2 + 2 = 4 \)
  - \( 4 + 2 = 6 \)
- **Type of Sequence:** Arithmetic (common difference = 2)

### C: \( -2, -6, -18, -54, \underline{\ \ }, \underline{\ \ }, \dots \)
- **Describe Pattern:** Each number is multiplied by -3.
  - \( -2 \times 3 = -6 \)
  - \( -6 \times 3 = -18 \)
  - \( -18 \times 3 = -54 \)
- The next terms would be:
  - \( -54 \times 3 = -162 \)
  - \( -162 \times 3 = -486 \)
- **Type of Sequence:** Geometric (ratio = 3)

### D: \( 2, 5, 10, 17, \underline{\ \ }, \underline{\ \ }, \dots \)
- **Describe Pattern:** The differences between consecutive numbers increase by 2.
  - \( 5 - 2 = 3 \)
  - \( 10 - 5 = 5 \)
  - \( 17 - 10 = 7 \)
- The next terms would be:
  - \( 17 + 9 = 26 \)
  - \( 26 + 11 = 37 \)
- **Type of Sequence:** Neither arithmetic nor geometric (quadratic)

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### E: \( 4, \frac{7}{4}, \frac{1}{2}, \frac{-11}{4}, \underline{\ \ }, \underline{\ \ }, \dots \)
- **Describe Pattern:** The differences between consecutive terms are decreasing by \( \frac{5}{4} \).
  - \( \frac{7}{4} - 4 = \frac{-9}{4} \)
  - \( \frac{1}{2} - \frac{7}{4} = \frac{-5}{4} \)
- The next term would be:
  - \( \frac{-11}{4} - \frac{-9}{4} = \frac{-19}{4} \)
- **Type of Sequence:** Arithmetic (common difference = \( -\frac{5}{4} \))

### F: \( 1234, 123.4, 12.34, 1.234, \underline{\ \ }, \underline{\ \ }, \dots \)
- **Describe Pattern:** Each term is divided by 10.
  - \( 1234 \div 10 = 123.4 \)
  - \( 123.4 \div 10 = 12.34 \)
  - \( 12.34 \div 10 = 1.234 \)
- The next terms would be:
  - \( 1.234 \div 10 = 0.1234 \)
  - \( 0.1234 \div 10 = 0.01234 \)
- **Type of Sequence:** Geometric (ratio = \( \frac{1}{10} \))

### G: \( 1, -2, 3, -4, 5, \underline{\ \ }, \underline{\ \ }, \dots \)
- **Describe Pattern:** The numbers alternate in sign and increase by 1 in absolute value.
  - \( 1, -2, 3, -4, 5 \)
- The next terms would be:
  - \( -6, 7 \)
- **Type of Sequence:** Neither arithmetic nor geometric (alternating pattern)

### H: \( -20, -16, -12, -8, -4, \underline{\ \ }, \underline{\ \ }, \dots \)
- **Describe Pattern:** The numbers are increasing by 4 each time.
  - \( -20 + 4 = -16 \)
  - \( -16 + 4 = -12 \)
  - \( -12 + 4 = -8 \)
  - \( -8 + 4 = -4 \)
- The next term would be:
  - \( -4 + 4 = 0 \)
- **Type of Sequence:** Arithmetic (common difference = 4)

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Would you like further explanation on any of these sequences?

Here are 5 additional questions to expand on the topic:
1. How do you differentiate between arithmetic and geometric sequences?
2. Can a sequence be both arithmetic and geometric?
3. What is the general formula for the \(n\)-th term of an arithmetic sequence?
4. How would you find the sum of the first \(n\) terms of a geometric sequence?
5. Can you determine the pattern of a sequence that is neither arithmetic nor geometric?

**Tip:** Recognizing patterns quickly in sequences helps in identifying the type of sequence and predicting future terms effectively.

Various sequence problems involving arithmetic and geometric patterns, requiring the user to identify the next terms and the type of sequence.

Explore sequence problems, including arithmetic and geometric patterns. Learn how to identify the next terms in a sequence and classify them as arithmetic or geometric, with detailed explanations. Suitable for students in Grades 6-8.

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