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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)

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)

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.