🔢 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:

1. Find next: 7, 14, 21, 28, 35, ?
2. Find missing: 3, 6, 12, 24, ?, 96
3. Find next: 1, 4, 9, 16, 25, ?

Level 2:

4. Find missing: 5, 11, 23, 47, ?, 191
5. Find next: 2, 5, 11, 20, 32, ?
6. Find wrong: 3, 5, 11, 29, 83, 256

Level 3:

7. Find next: 1, 1, 2, 6, 24, ?
8. Find missing: 3, 8, 5, 10, 7, 12, ?, 14
9. 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:

• Number Series Questions
• Shortcuts & Tricks

Master Number Series - Pattern recognition comes with practice! 🔢