Number Sequences
Understanding Mathematical Sequences
Number sequences are ordered lists of numbers that follow specific patterns or rules. Understanding these patterns is essential for mathematics, computer science, and logical reasoning. Each sequence type has unique properties and applications.
Arithmetic Sequences
Each term increases by a constant difference (common difference).
2, 5, 8, 11, 14, ...
Basic Arithmetic Sequence
Pattern: Add 3 to each term
Formula: an = 2 + (n-1) × 3
Next terms: 17, 20, 23
10, 7, 4, 1, -2, ...
Decreasing Arithmetic
Pattern: Subtract 3 from each term
Formula: an = 10 + (n-1) × (-3)
Next terms: -5, -8, -11
1, 1.5, 2, 2.5, 3, ...
Decimal Arithmetic
Pattern: Add 0.5 to each term
Formula: an = 1 + (n-1) × 0.5
Next terms: 3.5, 4, 4.5
Geometric Sequences
Each term is multiplied by a constant ratio (common ratio).
2, 6, 18, 54, 162, ...
Basic Geometric Sequence
Pattern: Multiply each term by 3
Formula: an = 2 × 3(n-1)
Next terms: 486, 1458, 4374
80, 40, 20, 10, 5, ...
Decreasing Geometric
Pattern: Multiply each term by 0.5
Formula: an = 80 × (0.5)(n-1)
Next terms: 2.5, 1.25, 0.625
1, -2, 4, -8, 16, ...
Alternating Geometric
Pattern: Multiply each term by -2
Formula: an = 1 × (-2)(n-1)
Next terms: -32, 64, -128
Quadratic Sequences
The second difference between terms is constant. Pattern involves n².
1, 4, 9, 16, 25, ...
Perfect Squares
Pattern: n² where n = 1, 2, 3, 4, 5...
Formula: an = n²
Next terms: 36, 49, 64
2, 8, 18, 32, 50, ...
Modified Quadratic
Pattern: 2n² where n = 1, 2, 3, 4, 5...
Formula: an = 2n²
Next terms: 72, 98, 128
3, 7, 13, 21, 31, ...
Complex Quadratic
Pattern: n² + n + 1
Formula: an = n² + n + 1
Next terms: 43, 57, 73
Special Famous Sequences
1, 1, 2, 3, 5, 8, 13, ...
Fibonacci Sequence
Pattern: Each term is the sum of the two preceding terms
Formula: Fn = Fn-1 + Fn-2
Next terms: 21, 34, 55
Applications: Nature patterns, golden ratio, computer algorithms
2, 3, 5, 7, 11, 13, 17, ...
Prime Numbers
Pattern: Numbers divisible only by 1 and themselves
No simple formula - must be calculated or tested
Next terms: 19, 23, 29
Applications: Cryptography, computer science, number theory
1, 3, 6, 10, 15, 21, ...
Triangular Numbers
Pattern: Sum of first n natural numbers
Formula: Tn = n(n+1)/2
Next terms: 28, 36, 45
Applications: Combinatorics, geometric arrangements
Problem-Solving Strategies
Step 1: Look for Patterns
Calculate differences between consecutive terms. If constant, it's arithmetic. If differences have a pattern, look deeper.
Step 2: Check Ratios
Divide each term by the previous term. If constant, it's geometric. Look for multiplication patterns.
Step 3: Test Formulas
Try n², n³, or combinations like an² + bn + c. Verify your formula works for all given terms.
Step 4: Consider Special Cases
Check if it matches famous sequences like Fibonacci, primes, or triangular numbers.
Practice Problems
Problem 1: Find the next three terms
4, 9, 16, 25, 36, ?, ?, ?
Show Solution
Answer: 49, 64, 81
Pattern: Perfect squares starting from 2²: (n+1)² where n starts at 1
Problem 2: Find the pattern
3, 7, 15, 31, 63, ?, ?, ?
Show Solution
Answer: 127, 255, 511
Pattern: 2ⁿ⁺¹ - 1 or each term = 2 × previous term + 1
Problem 3: Identify the sequence type
5, 20, 80, 320, 1280, ?, ?, ?
Show Solution
Answer: 5120, 20480, 81920
Pattern: Geometric sequence with ratio 4 (multiply by 4 each time)
Mastering Number Sequences
- Practice regularly - Work through different sequence types daily to build pattern recognition
- Show your work - Always write down the differences, ratios, or formulas you test
- Verify answers - Check that your formula produces all the given terms correctly
- Learn the classics - Memorize patterns for squares, cubes, Fibonacci, and triangular numbers
- Think systematically - Follow the problem-solving steps: differences → ratios → formulas → special cases
- Use technology - Graphing calculators can help visualize sequence patterns