Sequence Calculator

Generate and analyze mathematical sequences

Calculate arithmetic, geometric, Fibonacci, and custom sequences. Find nth terms, generate sequence values, and analyze mathematical patterns.

Enter the starting value for the sequence

Maximum 100 terms for optimal performance

Find the value of the sequence at position n

Sequence Examples

Click on any example to explore different types of mathematical sequences

Arithmetic Sequence

arithmetic

Even numbers sequence with common difference 2

First Term: 2

Common Difference: 2

Number of Terms: 10

Geometric Sequence

geometric

Powers of 3 with common ratio 3

First Term: 3

Common Ratio: 3

Number of Terms: 8

Fibonacci Sequence

fibonacci

Classic Fibonacci sequence starting with 0, 1

Number of Terms: 15

Square Numbers

custom

Perfect squares sequence

Custom Terms: 1, 4, 9, 16, 25, 36, 49

Other Titles
Understanding Sequences: A Comprehensive Guide
Master the fundamentals of mathematical sequences, their types, properties, and real-world applications

What are Mathematical Sequences?

  • Definition and Basic Concepts
  • Types of Sequences
  • Sequence Notation and Terminology
A mathematical sequence is an ordered list of numbers that follow a specific pattern or rule. Each number in the sequence is called a term, and the position of each term is indicated by its index. Sequences are fundamental mathematical objects that appear throughout mathematics, from basic arithmetic to advanced analysis.
Key Characteristics of Sequences
Every sequence has several important characteristics: the order of terms matters, each term has a specific position (index), and there's usually a pattern or rule that determines how terms are generated. Understanding these characteristics is essential for working with sequences effectively.
Common Types of Sequences
The most important types of sequences include arithmetic sequences (constant difference), geometric sequences (constant ratio), Fibonacci sequences (sum of previous two terms), and many other special sequences like prime numbers, perfect squares, and factorial sequences.

Basic Sequence Examples

  • 2, 4, 6, 8, 10... (arithmetic)
  • 1, 2, 4, 8, 16... (geometric)
  • 1, 1, 2, 3, 5, 8... (Fibonacci)

Arithmetic Sequences

  • Definition and Properties
  • General Term Formula
  • Sum Calculations
An arithmetic sequence is a sequence where each term after the first is obtained by adding a constant value called the common difference (d) to the previous term. This creates a linear pattern that's easy to predict and analyze.
General Term Formula
The nth term of an arithmetic sequence is given by: a(n) = a₁ + (n-1)d, where a₁ is the first term, d is the common difference, and n is the term position. This formula allows you to find any term without calculating all previous terms.
Sum of Arithmetic Sequences
The sum of the first n terms of an arithmetic sequence is: S(n) = n/2 × [2a₁ + (n-1)d] or S(n) = n/2 × (first term + last term). This formula is derived using the method of pairing terms from opposite ends of the sequence.

Arithmetic Sequence Examples

  • 3, 7, 11, 15, 19... (first term = 3, common difference = 4)
  • 10, 5, 0, -5, -10... (first term = 10, common difference = -5)

Geometric Sequences

  • Definition and Properties
  • General Term Formula
  • Convergence and Divergence
A geometric sequence is a sequence where each term after the first is obtained by multiplying the previous term by a constant value called the common ratio (r). This creates exponential growth or decay patterns.
General Term Formula
The nth term of a geometric sequence is given by: a(n) = a₁ × r^(n-1), where a₁ is the first term, r is the common ratio, and n is the term position. This exponential formula shows how quickly geometric sequences can grow or shrink.
Sum of Geometric Sequences
The sum of the first n terms of a geometric sequence is: S(n) = a₁ × (1 - r^n) / (1 - r) for r ≠ 1, or S(n) = n × a₁ for r = 1. For infinite geometric series with |r| < 1, the sum converges to a₁ / (1 - r).

Geometric Sequence Examples

  • 2, 6, 18, 54, 162... (first term = 2, common ratio = 3)
  • 64, 32, 16, 8, 4... (first term = 64, common ratio = 1/2)

Real-World Applications of Sequences

  • Financial Applications
  • Scientific and Engineering Uses
  • Computer Science Applications
Sequences appear throughout real-world applications, from financial calculations to scientific modeling. Understanding sequence properties helps solve practical problems in various fields.
Financial Applications
Arithmetic sequences model regular savings plans, loan payments, and salary progressions. Geometric sequences model compound interest, population growth, and depreciation calculations. These applications make sequences essential for financial planning and analysis.
Scientific and Engineering Applications
In physics, sequences model wave patterns, radioactive decay, and oscillations. In engineering, they appear in signal processing, control systems, and optimization algorithms. Scientists use sequences in data analysis, experimental design, and mathematical modeling.

Real-World Sequence Applications

  • Monthly savings: $100, $200, $300... (arithmetic)
  • Compound interest: $1000, $1050, $1102.50... (geometric)
  • Fibonacci in nature: flower petals, pine cones, shells

Advanced Sequence Analysis

  • Sequence Properties and Behavior
  • Convergence and Limits
  • Pattern Recognition Techniques
Advanced sequence analysis involves studying convergence, limits, and complex patterns. This deeper understanding is crucial for calculus, analysis, and advanced mathematical applications.
Convergence and Limits
Many sequences approach a specific value (limit) as n approaches infinity. Understanding convergence helps determine long-term behavior and is fundamental to calculus and mathematical analysis. Convergent sequences have practical applications in numerical methods and approximation techniques.
Pattern Recognition
Identifying sequence patterns involves analyzing differences, ratios, and recursive relationships. Modern computational tools can detect complex patterns that might not be obvious through manual inspection, enabling discovery of new mathematical relationships and properties.

Advanced Sequence Concepts

  • 1/n → 0 as n → ∞ (convergent)
  • (-1)^n oscillates between -1 and 1 (divergent)
  • Recursive: a(n) = 2a(n-1) + 1 with a(1) = 1