🔢 Number Series - Formula Sheet

🎯 Basic Patterns

Arithmetic Progression (AP)

Common difference (d) constant
Formula: a, a+d, a+2d, a+3d, ...
Example: 3, 7, 11, 15, 19 (d=4)

Geometric Progression (GP)

Common ratio (r) constant
Formula: a, ar, ar², ar³, ...
Example: 2, 6, 18, 54, 162 (r=3)

Squares/Cubes

Perfect squares: 1, 4, 9, 16, 25
Perfect cubes: 1, 8, 27, 64, 125
Square + 1: 2, 5, 10, 17, 26
Cube - 1: 0, 7, 26, 63, 124

📊 Advanced Patterns

Alternating Series

Two interleaved patterns
Example: 2, 8, 4, 16, 6, 24
Pattern 1: 2, 4, 6 (+2)
Pattern 2: 8, 16, 24 (+8)

Product/Sum Patterns

Multiplication: 2, 6, 18, 54 (×3)
Addition + Multiplication: 2, 4, 12, 48 (+2, ×3, ×4)
Square operations: 2, 4, 16, 256 (×2, ×4, ×16)

Digit Operations

Digit sum: 23, 5, 14, 5, 17, 8
Digit product: 34, 12, 26, 12, 39, 27
Alternating digits: 1234, 13, 24, 37, 11

🔢 Special Series

Fibonacci Series

Sum of previous two terms
Example: 0, 1, 1, 2, 3, 5, 8, 13

Prime Numbers

2, 3, 5, 7, 11, 13, 17, 19, 23

Factorial Series

1!, 2!, 3!, 4!, 5!
1, 2, 6, 24, 120

Triangular Numbers

1, 3, 6, 10, 15, 21
Formula: n(n+1)/2

⚡ Complex Patterns

Multiple Operations

Pattern: ×2+1, ×2+2, ×2+3, ...
Example: 3, 7, 16, 35, 72
3×2+1=7, 7×2+2=16, 16×2+3=35, 35×2+2=72

Position-based

Square positions: 1², 2², 3², 4²
Cube positions: 1³, 2³, 3³, 4³
Prime positions: 2nd, 3rd, 5th, 7th terms

Difference Patterns

Second order difference constant:
Original: 2, 5, 10, 17, 26
1st diff: 3, 5, 7, 9
2nd diff: 2, 2, 2 (constant)

📝 Solving Strategy

Step-by-Step Approach

1. Check basic patterns (AP, GP)
2. Look for squares/cubes
3. Check alternating patterns
4. Examine differences between terms
5. Try product/sum patterns
6. Consider digit operations
7. Verify pattern throughout

Quick Checks

Differences: Constant or pattern?
Ratios: Constant or pattern?
Special numbers: Primes, squares, cubes?
Alternating: Two separate patterns?

🔍 Common Question Types

Missing Number

Find the missing term in sequence
Example: 2, 5, ?, 17, 26 (Answer: 10)

Wrong Number

Identify number that doesn't fit pattern
Example: 2, 4, 8, 17, 32 (17 is wrong, should be 16)

Next Number

Find the next term in sequence
Example: 3, 7, 15, 31, ? (Answer: 63)

⚡ Quick Tips

Pattern Recognition

1. Always check simple patterns first
2. Calculate differences between terms
3. Look for multiplication patterns
4. Consider digit-based patterns
5. Check for alternating patterns

Common Mistakes to Avoid

Don't assume pattern too early
Check pattern with all given terms
Consider multiple possibilities
Verify answer fits complete pattern

Time Management

1. Quick scan for obvious patterns
2. Systematic checking of common patterns
3. Skip very complex patterns initially
4. Return if time permits

📚 Practice Areas

Mental Calculation

Quick multiplication/division
Square and cube calculations
Digit sum operations

Pattern Recognition

Common mathematical sequences
Visual pattern identification
Logical reasoning skills

Master Number Series - Pattern, verify, apply! 🔢