Number Series - Theory & Concepts
🔢 Number Series - Complete Theory
Master pattern recognition - the fastest scoring topic in IBPS!
🎯 What is Number Series?
Number Series is a sequence of numbers following a specific pattern or rule.
Your Task: Find the missing number or next number in the sequence.
Key Skill: Pattern Recognition
📊 Types of Series
1. Difference Series
Same Difference (Arithmetic Progression):
Pattern: Each term = Previous term + constant
Example: 5, 8, 11, 14, 17, ? Difference: +3, +3, +3, +3 Next: 17 + 3 = 20
Increasing Difference:
Example: 2, 3, 5, 8, 12, ? Difference: +1, +2, +3, +4 Next: 12 + 5 = 17
Decreasing Difference:
Example: 20, 19, 17, 14, 10, ? Difference: -1, -2, -3, -4 Next: 10 - 5 = 5
2. Ratio Series (Geometric Progression)
Pattern: Each term = Previous term × constant
Example: 3, 6, 12, 24, 48, ? Ratio: ×2, ×2, ×2, ×2 Next: 48 × 2 = 96
Alternating Ratio:
Example: 2, 6, 18, 54, ? Ratio: ×3, ×3, ×3 Next: 54 × 3 = 162
3. Square Series
Based on squares: n², (n+1)², (n+2)², …
Example: 1, 4, 9, 16, 25, ? Pattern: 1², 2², 3², 4², 5² Next: 6² = 36
Square + Constant:
Example: 2, 5, 10, 17, 26, ? Pattern: 1²+1, 2²+1, 3²+1, 4²+1, 5²+1 Next: 6²+1 = 37
Square - Constant:
Example: 0, 3, 8, 15, 24, ? Pattern: 1²-1, 2²-1, 3²-1, 4²-1, 5²-1 Next: 6²-1 = 35
4. Cube Series
Based on cubes: n³, (n+1)³, (n+2)³, …
Example: 1, 8, 27, 64, 125, ? Pattern: 1³, 2³, 3³, 4³, 5³ Next: 6³ = 216
Cube ± Constant:
Example: 9, 28, 65, 126, 217, ? Pattern: 2³+1, 3³+1, 4³+1, 5³+1, 6³+1 Next: 7³+1 = 344
5. Prime Number Series
Example: 2, 3, 5, 7, 11, 13, ? Pattern: Prime numbers Next: 17
Example: 4, 9, 25, 49, 121, ? Pattern: Squares of primes (2², 3², 5², 7², 11²) Next: 13² = 169
6. Two-Tier Series
Alternating Operations:
Example: 2, 5, 4, 7, 6, 9, 8, ?
Odd positions: 2, 4, 6, 8 (+2 series) Even positions: 5, 7, 9 (+2 series)
Next is even position: 9 + 2 = 11
Two Different Patterns:
Example: 1, 2, 4, 7, 11, 16, 22, ?
Differences: +1, +2, +3, +4, +5, +6 Next: 22 + 7 = 29
7. Mixed Operations Series
Example: 2, 5, 11, 23, 47, ? Pattern: ×2 +1, ×2 +1, ×2 +1, ×2 +1
2 → 2×2+1 = 5 5 → 5×2+1 = 11 11 → 11×2+1 = 23 23 → 23×2+1 = 47 47 → 47×2+1 = 95
8. Fibonacci-Type Series
Each term = Sum of previous two terms
Classic: 0, 1, 1, 2, 3, 5, 8, 13, ? Next: 8 + 13 = 21
Modified: 1, 3, 4, 7, 11, 18, ? Next: 11 + 18 = 29
💡 Solved Examples
Example 1: Simple Difference
Q: Find missing: 12, 17, 22, 27, ?, 37
Solution:
Differences: +5, +5, +5, +5 Missing = 27 + 5 = 32
Answer: 32
Example 2: Increasing Difference
Q: Find missing: 3, 5, 8, 12, 17, ?
Solution:
Differences: +2, +3, +4, +5 Next difference: +6 Missing = 17 + 6 = 23
Answer: 23
Example 3: Ratio Pattern
Q: Find wrong: 5, 10, 20, 40, 85, 160
Solution:
Check ratios: ×2, ×2, ×2, ×2, ×2 5 → 10 (×2) ✓ 10 → 20 (×2) ✓ 20 → 40 (×2) ✓ 40 → 80 (×2) should be 80, not 85! 80 → 160 (×2) ✓
Wrong number: 85
Answer: 85
Example 4: Square Series
Q: Find missing: 2, 5, 10, 17, ?, 37
Solution:
Check pattern: 2 = 1² + 1 5 = 2² + 1 10 = 3² + 1 17 = 4² + 1 ? = 5² + 1 = 26 37 = 6² + 1 ✓
Answer: 26
Example 5: Alternating Series
Q: Find missing: 3, 8, 5, 10, 7, 12, ?, 14
Solution:
Odd positions: 3, 5, 7, ? → +2 each Even positions: 8, 10, 12, 14 → +2 each
Missing is 7th (odd position): 7 + 2 = 9
Answer: 9
Example 6: Mixed Operations
Q: Find missing: 4, 9, 19, 39, 79, ?
Solution:
4 → 4×2+1 = 9 9 → 9×2+1 = 19 19 → 19×2+1 = 39 39 → 39×2+1 = 79 79 → 79×2+1 = 159
Answer: 159
Example 7: Fibonacci Type
Q: Find missing: 2, 3, 5, 8, ?, 21
Solution:
Each = Sum of previous two 2 + 3 = 5 3 + 5 = 8 5 + 8 = 13 8 + 13 = 21 ✓
Answer: 13
Example 8: Cube Pattern
Q: Find wrong: 28, 65, 126, 215, 344
Solution:
Check cube pattern: 28 = 3³ + 1 ✓ 65 = 4³ + 1 ✓ 126 = 5³ + 1 ✓ 215 = 6³ - 1 (should be 6³ + 1 = 217) 344 = 7³ + 1 ✓
Wrong: 215
Answer: 215
⚡ Quick Pattern Recognition
Step-by-Step Approach
Step 1: Check Simple Difference
Calculate differences between consecutive terms If constant → Arithmetic series If increasing/decreasing → Look at 2nd level differences
Step 2: Check Ratio
Divide each term by previous If constant → Geometric series
Step 3: Check Squares/Cubes
Does √term give sequence? Does ∛term give sequence? Check term ± small number
Step 4: Check Alternating Pattern
Separate odd and even positions Look for pattern in each
Step 5: Check Mixed Operations
Try ×2+1, ×2-1, ×3+2, etc.
📊 Common Patterns (Memorize!)
Perfect Squares (1-20)
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400
Perfect Cubes (1-10)
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
Prime Numbers (1-100)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
Fibonacci Sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…
🎯 Advanced Patterns
1. Digit Sum Pattern
Example: 11, 13, 17, 23, 31, ? Digit sums: 2, 4, 8, 5, 4 Look for pattern in digit sums
2. Reverse Operations
Example: 96, 48, 24, 12, 6, ? Pattern: ÷2, ÷2, ÷2, ÷2 Next: 6 ÷ 2 = 3
3. Combined Series
Example: 1, 1, 2, 6, 24, ? Pattern: 1×1, 1×2, 2×3, 6×4 Next: 24×5 = 120 (factorial pattern!)
4. Gap Series
Example: 2, ?, 18, ?, 50, ? Position pattern based on gaps
⚠️ Common Mistakes
❌ Mistake 1: Assuming Simple Pattern
Wrong: Always checking only +/- patterns ✗ Right: Try multiple approaches ✓
❌ Mistake 2: Ignoring Alternating Patterns
Wrong: Looking at all terms together ✗ Right: Separate odd/even positions ✓
❌ Mistake 3: Not Checking Wrong Number Type
In “find wrong number”, wrong one often breaks pattern slightly Check if all others follow pattern!
❌ Mistake 4: Calculation Errors
Wrong: 47 × 2 + 1 = 94 ✗ Right: 47 × 2 + 1 = 95 ✓
⚡ Speed Tricks
Trick 1: First Three Terms
Usually pattern is clear from first 3-4 terms Focus on these first!
Trick 2: Difference of Differences
If first difference unclear: Calculate difference of differences (2nd level)
Example: 2, 3, 5, 8, 12 Diff: 1, 2, 3, 4 (clear pattern!)
Trick 3: Eliminate Options
In MCQs, eliminate obviously wrong options Check if remaining options follow pattern
Trick 4: Work Backwards
If finding missing in middle: Check pattern from both directions
📝 Practice Problems
Level 1:
- Find next: 7, 14, 21, 28, 35, ?
- Find missing: 3, 6, 12, 24, ?, 96
- Find next: 1, 4, 9, 16, 25, ?
Level 2:
- Find missing: 5, 11, 23, 47, ?, 191
- Find next: 2, 5, 11, 20, 32, ?
- Find wrong: 3, 5, 11, 29, 83, 256
Level 3:
- Find next: 1, 1, 2, 6, 24, ?
- Find missing: 3, 8, 5, 10, 7, 12, ?, 14
- Find wrong: 28, 65, 126, 215, 344, 513
🔗 Related Topics
Prerequisites:
- Number System - Squares, cubes, primes
- Simplification - Quick calculations
Related:
- Average - Series sums
- Percentage - Percentage-based series
Practice:
🎯 Continue Your Learning Journey
Master Number Series - Pattern recognition comes with practice! 🔢
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