5, 9, 13, 17, __, __, __
Arithmetic sequence: A sequence where each term increases by the same amount (common difference)
- Find the difference between consecutive terms
- Identify the pattern rule
- Apply the rule to find missing terms
9 - 5 = 4, 13 - 9 = 4, 17 - 13 = 4
The common difference is 4
Add 4 to get the next number in the sequence
Pattern rule: Start at 5, add 4 each time
17 + 4 = 21
21 + 4 = 25
25 + 4 = 29
5 → 9 → 13 → 17 → 21 → 25 → 29
Each number increases by 4 ✓
The next 3 numbers are: 21, 25, 29
• Arithmetic sequence: Each term differs by a constant value
• Difference calculation: Subtract consecutive terms to find the pattern
• Pattern continuation: Apply the rule consistently
△, ○, □, △, ○, □, __
Geometric pattern: A repeating sequence of shapes, colors, or symbols
Look at the sequence: △, ○, □, △, ○, □
The pattern repeats every 3 shapes: △, ○, □
Triangle (△), Circle (○), Square (□) repeats
After □ comes △ again
Since the last shape is □, the next shape follows the cycle
After □ comes △
△, ○, □, △, ○, □, △
Pattern continues correctly ✓
The next shape is: △ (Triangle)
• Repeating pattern: Identify the smallest repeating unit
• Cycle recognition: Count how many elements make up one cycle
• Continuation: Continue the established cycle
2, 6, 18, 54, __, __
Multiplication pattern: Each term is multiplied by the same number to get the next term
6 ÷ 2 = 3
18 ÷ 6 = 3
54 ÷ 18 = 3
Each term is multiplied by 3
Multiply each term by 3 to get the next term
Pattern rule: Start at 2, multiply by 3 each time
54 × 3 = 162
162 × 3 = 486
2 → 6 → 18 → 54 → 162 → 486
Each term is 3 times the previous term ✓
The next 2 numbers are: 162, 486
• Multiplication pattern: Each term is multiplied by a constant factor
• Factor identification: Divide consecutive terms to find the multiplier
• Consistency: Apply the same operation throughout
Pattern: A regular, repeated sequence following a specific rule
Arithmetic sequence: Numbers increasing by a constant difference
Geometric sequence: Numbers multiplied by a constant factor
- Observe the sequence: Look at the first few terms carefully
- Find the relationship: Check if adding, subtracting, multiplying, or dividing gives the next term
- Test the rule: Apply the rule to multiple pairs to confirm
- Apply the rule: Use the rule to find missing terms
- Verify: Check that the completed pattern makes sense
• Arithmetic sequence: \(a_n = a_1 + (n-1)d\) where d is the common difference
• Geometric sequence: \(a_n = a_1 \times r^{n-1}\) where r is the common ratio
• Repeating pattern: Identify the cycle length and apply modulo operation
• Pattern verification: Always check that your rule works for multiple consecutive terms
3, 6, 12, 24, 48, __, __
Geometric sequence: Each term is multiplied by the same number (common ratio)
6 ÷ 3 = 2
12 ÷ 6 = 2
24 ÷ 12 = 2
48 ÷ 24 = 2
The common ratio is 2
Multiply each term by 2 to get the next term
Pattern rule: Start at 3, multiply by 2 each time
5th term: 48
6th term: 48 × 2 = 96
7th term: 96 × 2 = 192
3 → 6 → 12 → 24 → 48 → 96 → 192
Each term is 2 times the previous term ✓
The 7th term is: 192
• Geometric sequence: Each term is multiplied by the same constant
• Ratio identification: Divide consecutive terms to find the common ratio
• Term calculation: Continue applying the rule to find specific terms
1, 4, 9, 16, __, 36, 49
Square number pattern: Each term is a perfect square (n²)
1 = 1²
4 = 2²
9 = 3²
16 = 4²
36 = 6²
49 = 7²
The sequence shows consecutive perfect squares
Pattern rule: n² where n starts at 1 and increases by 1 each time
Position 1: 1² = 1
Position 2: 2² = 4
Position 3: 3² = 9
Position 4: 4² = 16
Position 5: 5² = 25
Position 6: 6² = 36
Position 7: 7² = 49
1 → 4 → 9 → 16 → 25 → 36 → 49
Which is: 1² → 2² → 3² → 4² → 5² → 6² → 7² ✓
The missing number is: 25
• Square number pattern: Recognize perfect squares
• Position analysis: Consider the position of each term
• Verification: Ensure the identified rule works for all known terms
Pattern: A predictable sequence following a specific rule
Arithmetic sequence: Numbers with a constant difference between consecutive terms
Geometric sequence: Numbers with a constant ratio between consecutive terms
Perfect square: A number that can be expressed as n² where n is a whole number
- Observe: Carefully examine the given sequence
- Compare: Look for relationships between consecutive terms
- Test: Apply potential rules to multiple pairs
- Confirm: Verify the rule works consistently
- Apply: Use the rule to find missing terms
- Validate: Check the final answer fits the pattern
• Arithmetic sequence: \(a_n = a_1 + (n-1)d\) where d is the common difference
• Geometric sequence: \(a_n = a_1 \times r^{n-1}\) where r is the common ratio
• Perfect squares: 1²=1, 2²=4, 3²=9, 4²=16, 5²=25, 6²=36, 7²=49, 8²=64, 9²=81, 10²=100
• Repeating pattern: Use division with remainder to find position in cycle
Questions & Answers
Question: I have trouble knowing when to add vs multiply in pattern problems. How can I tell the difference?
Answer: Great question! Here's how to distinguish between addition and multiplication patterns:
- Addition patterns (arithmetic): Look for a constant difference between consecutive terms. Example: 3, 7, 11, 15... (difference is 4)
- Multiplication patterns (geometric): Look for a constant ratio between consecutive terms. Example: 2, 6, 18, 54... (ratio is 3)
To determine which pattern you have:
- Calculate the difference: next - previous (for addition patterns)
- Calculate the ratio: next ÷ previous (for multiplication patterns)
- If differences are the same, it's addition. If ratios are the same, it's multiplication.
Example: For sequence 4, 8, 12, 16...
- Differences: 8-4=4, 12-8=4, 16-12=4 (same difference = addition pattern)
- Ratios: 8÷4=2, 12÷8=1.5, 16÷12≈1.33 (different ratios = not multiplication pattern)
Question: How do I solve patterns with shapes instead of numbers? I don't understand the rules.
Answer: Shape patterns follow the same principles as number patterns! Here's how to approach them:
- Identify the cycle: Look for the smallest repeating group of shapes
- Count the cycle length: How many shapes make up one complete repetition?
- Continue the pattern: Repeat the same sequence of shapes
Example: Triangle, Circle, Square, Triangle, Circle, Square...
- The cycle is: Triangle, Circle, Square (length = 3)
- After Square comes Triangle again
- Pattern continues: Triangle, Circle, Square, Triangle, Circle, Square, Triangle...
Tip: Draw the pattern if needed, or write the shapes in order to visualize the repeating unit!
Question: What should I do when a pattern has two different rules mixed together? I got confused on a problem like 2, 4, 8, 10, 20, 22...
Answer: Some complex patterns have alternating rules! Let's analyze your example: 2, 4, 8, 10, 20, 22...
Looking closely:
- 2 → 4: Added 2
- 4 → 8: Multiplied by 2
- 8 → 10: Added 2
- 10 → 20: Multiplied by 2
- 20 → 22: Added 2
The pattern alternates: "Add 2, then Multiply by 2"
The next terms would be: 22 × 2 = 44, then 44 + 2 = 46
When facing complex patterns:
- Check if it's arithmetic or geometric first
- If not, look for alternating rules
- Consider if odd/even positions follow different rules
- Write out the operations to see the pattern clearly