Number Patterns worksheet for practicing sequence extension and pattern recognition.

Problem Overview:

The task involves extending number patterns and identifying the rule governing each sequence. The goal is to determine the next two numbers in each sequence and describe the pattern rule.

Solution Approach:

1. Analyze Each Sequence: Look for a consistent mathematical operation (e.g., addition, subtraction, multiplication, division) or a combination of operations.
2. Extend the Pattern: Use the identified rule to calculate the next two numbers in the sequence.
3. State the Pattern Rule: Clearly describe the rule that governs the sequence.

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Detailed Solutions:

#### 1. Sequence: 4, 8, 11, 22, 25, 50, 53
- Pattern Analysis:
- From 4 to 8: \( 4 \times 2 = 8 \)
- From 8 to 11: \( 8 + 3 = 11 \)
- From 11 to 22: \( 11 \times 2 = 22 \)
- From 22 to 25: \( 22 + 3 = 25 \)
- From 25 to 50: \( 25 \times 2 = 50 \)
- From 50 to 53: \( 50 + 3 = 53 \)
- Rule: Multiply by 2, then add 3.
- Next Two Numbers:
- From 53: \( 53 \times 2 = 106 \)
- From 106: \( 106 + 3 = 109 \)
- Answer: 106, 109

#### 2. Sequence: 20, 15, 25, 20, 30, 25, 35
- Pattern Analysis:
- From 20 to 15: \( 20 - 5 = 15 \)
- From 15 to 25: \( 15 + 10 = 25 \)
- From 25 to 20: \( 25 - 5 = 20 \)
- From 20 to 30: \( 20 + 10 = 30 \)
- From 30 to 25: \( 30 - 5 = 25 \)
- From 25 to 35: \( 25 + 10 = 35 \)
- Rule: Subtract 5, then add 10.
- Next Two Numbers:
- From 35: \( 35 - 5 = 30 \)
- From 30: \( 30 + 10 = 40 \)
- Answer: 30, 40

#### 3. Sequence: 48, 57, 65, 72, 78, 83
- Pattern Analysis:
- From 48 to 57: \( 57 - 48 = 9 \)
- From 57 to 65: \( 65 - 57 = 8 \)
- From 65 to 72: \( 72 - 65 = 7 \)
- From 72 to 78: \( 78 - 72 = 6 \)
- From 78 to 83: \( 83 - 78 = 5 \)
- Rule: The difference decreases by 1 each time.
- Next Two Numbers:
- From 83: \( 83 + 4 = 87 \)
- From 87: \( 87 + 3 = 90 \)
- Answer: 87, 90

#### 4. Sequence: 2, 4, 6, 12, 14, 28, 30
- Pattern Analysis:
- From 2 to 4: \( 2 \times 2 = 4 \)
- From 4 to 6: \( 4 + 2 = 6 \)
- From 6 to 12: \( 6 \times 2 = 12 \)
- From 12 to 14: \( 12 + 2 = 14 \)
- From 14 to 28: \( 14 \times 2 = 28 \)
- From 28 to 30: \( 28 + 2 = 30 \)
- Rule: Multiply by 2, then add 2.
- Next Two Numbers:
- From 30: \( 30 \times 2 = 60 \)
- From 60: \( 60 + 2 = 62 \)
- Answer: 60, 62

#### 5. Sequence: 3, 6, 5, 10, 9, 18, 17
- Pattern Analysis:
- From 3 to 6: \( 3 \times 2 = 6 \)
- From 6 to 5: \( 6 - 1 = 5 \)
- From 5 to 10: \( 5 \times 2 = 10 \)
- From 10 to 9: \( 10 - 1 = 9 \)
- From 9 to 18: \( 9 \times 2 = 18 \)
- From 18 to 17: \( 18 - 1 = 17 \)
- Rule: Multiply by 2, then subtract 1.
- Next Two Numbers:
- From 17: \( 17 \times 2 = 34 \)
- From 34: \( 34 - 1 = 33 \)
- Answer: 34, 33

#### 6. Sequence: 53, 50, 47, 44, 41, 38, 35, 32
- Pattern Analysis:
- From 53 to 50: \( 53 - 3 = 50 \)
- From 50 to 47: \( 50 - 3 = 47 \)
- From 47 to 44: \( 47 - 3 = 44 \)
- From 44 to 41: \( 44 - 3 = 41 \)
- From 41 to 38: \( 41 - 3 = 38 \)
- From 38 to 35: \( 38 - 3 = 35 \)
- From 35 to 32: \( 35 - 3 = 32 \)
- Rule: Subtract 3.
- Next Two Numbers:
- From 32: \( 32 - 3 = 29 \)
- From 29: \( 29 - 3 = 26 \)
- Answer: 29, 26

#### 7. Sequence: 94, 91, 86, 79, 70, 59, 46
- Pattern Analysis:
- From 94 to 91: \( 94 - 3 = 91 \)
- From 91 to 86: \( 91 - 5 = 86 \)
- From 86 to 79: \( 86 - 7 = 79 \)
- From 79 to 70: \( 79 - 9 = 70 \)
- From 70 to 59: \( 70 - 11 = 59 \)
- From 59 to 46: \( 59 - 13 = 46 \)
- Rule: Subtract an increasing odd number (3, 5, 7, 9, 11, 13, ...).
- Next Two Numbers:
- From 46: \( 46 - 15 = 31 \)
- From 31: \( 31 - 17 = 14 \)
- Answer: 31, 14

#### 8. Sequence: 63, 67, 64, 68, 65, 69, 66
- Pattern Analysis:
- From 63 to 67: \( 63 + 4 = 67 \)
- From 67 to 64: \( 67 - 3 = 64 \)
- From 64 to 68: \( 64 + 4 = 68 \)
- From 68 to 65: \( 68 - 3 = 65 \)
- From 65 to 69: \( 65 + 4 = 69 \)
- From 69 to 66: \( 69 - 3 = 66 \)
- Rule: Add 4, then subtract 3.
- Next Two Numbers:
- From 66: \( 66 + 4 = 70 \)
- From 70: \( 70 - 3 = 67 \)
- Answer: 70, 67

#### 9. Sequence: 19, 25, 32, 40, 49, 59, 70
- Pattern Analysis:
- From 19 to 25: \( 25 - 19 = 6 \)
- From 25 to 32: \( 32 - 25 = 7 \)
- From 32 to 40: \( 40 - 32 = 8 \)
- From 40 to 49: \( 49 - 40 = 9 \)
- From 49 to 59: \( 59 - 49 = 10 \)
- From 59 to 70: \( 70 - 59 = 11 \)
- Rule: The difference increases by 1 each time.
- Next Two Numbers:
- From 70: \( 70 + 12 = 82 \)
- From 82: \( 82 + 13 = 95 \)
- Answer: 82, 95

#### 10. Sequence: 29, 31, 34, 38, 43, 49, 56
- Pattern Analysis:
- From 29 to 31: \( 31 - 29 = 2 \)
- From 31 to 34: \( 34 - 31 = 3 \)
- From 34 to 38: \( 38 - 34 = 4 \)
- From 38 to 43: \( 43 - 38 = 5 \)
- From 43 to 49: \( 49 - 43 = 6 \)
- From 49 to 56: \( 56 - 49 = 7 \)
- Rule: The difference increases by 1 each time.
- Next Two Numbers:
- From 56: \( 56 + 8 = 64 \)
- From 64: \( 64 + 9 = 73 \)
- Answer: 64, 73

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Final Answers:

1. 106, 109
2. 30, 40
3. 87, 90
4. 60, 62
5. 34, 33
6. 29, 26
7. 31, 14
8. 70, 67
9. 82, 95
10. 64, 73

\boxed{
\begin{array}{l}
1. 106, 109 \\
2. 30, 40 \\
3. 87, 90 \\
4. 60, 62 \\
5. 34, 33 \\
6. 29, 26 \\
7. 31, 14 \\
8. 70, 67 \\
9. 82, 95 \\
10. 64, 73 \\
\end{array}
}