Free online sequence calculator for arithmetic, geometric, and Fibonacci sequences. Generate terms, find patterns, and calculate series sums with step-by-step solutions.
Mathematical sequences are fundamental concepts in algebra and number theory that represent ordered lists of numbers following specific patterns or rules. Our sequence calculator helps you generate terms, identify patterns, and calculate sums for various types of sequences. Whether you're a student learning about arithmetic and geometric progressions or exploring the fascinating Fibonacci sequence, our tool provides comprehensive solutions with detailed explanations.
This powerful calculator supports multiple sequence types, including arithmetic sequences (constant difference), geometric sequences (constant ratio), and special sequences like Fibonacci, making it an invaluable resource for education and mathematical exploration.
The starting value of your sequence
The constant amount added to each term
How many terms to calculate (max: 100)
Formula for nth term: aₙ = a₁ + (n - 1) × d
Formula for sum: Sₙ = n/2 × (2a₁ + (n - 1) × d)
Where: a₁ = first term, d = common difference, n = number of terms
Finding the 15th term of the sequence 2, 5, 8, 11... where a₁=2 and d=3: a₁₅ = 2 + (15-1)×3 = 44
Finding the 8th term of the sequence 3, 6, 12, 24... where a₁=3 and r=2: a₈ = 3 × 2⁷ = 384
Generating the sequence where each term is the sum of the two preceding: 0, 1, 1, 2, 3, 5, 8, 13, 21...
Calculating the sum of the first 10 terms of the arithmetic sequence 1, 4, 7, 10...: S₁₀ = 10/2[2(1) + (10-1)3] = 145
Students and teachers exploring sequence concepts and solving homework problems
Calculating compound interest, annuities, and recurring payment schedules
Algorithm analysis, recursive function design, and data structure planning
A sequence is an ordered list of numbers called terms, which are generated according to a specific rule or pattern. Each term in a sequence has a position, and the relationship between terms determines the sequence type.
In an arithmetic sequence, each term is obtained by adding a constant value (common difference) to the previous term. The general form is: aₙ = a₁ + (n-1)d, where a₁ is the first term and d is the common difference.
In a geometric sequence, each term is obtained by multiplying the previous term by a constant value (common ratio). The general form is: aₙ = a₁ × r^(n-1), where a₁ is the first term and r is the common ratio.
The Fibonacci sequence is a special sequence where each term is the sum of the two preceding terms, starting with 0 and 1. It has numerous applications in nature, art, and computer science, showing the golden ratio properties.
A sequence is a list of numbers in a specific order, while a series is the sum of the terms of a sequence. For example, the sequence 1, 2, 3, 4 has the corresponding series 1+2+3+4=10.
For an arithmetic sequence, the difference between consecutive terms is constant. For a geometric sequence, the ratio between consecutive terms is constant. Check these relationships to identify the type.
Our calculator can identify formulas for arithmetic, geometric, and Fibonacci sequences. For more complex patterns, the sequence pattern finder tool can help determine if a simple formula exists.
The Fibonacci sequence appears in nature (flower petals, pine cone spirals), financial markets (Fibonacci retracements), computer algorithms, and has connections to the golden ratio, making it important in art and architecture.
An infinite geometric series converges and has a finite sum only if the common ratio r satisfies |r| < 1. The sum is calculated as S = a₁/(1-r), where a₁ is the first term.
While this calculator focuses on fundamental sequence types, it provides a solid foundation for understanding more advanced mathematical concepts involving sequences and series in calculus and analysis.