Number Sequence Calculator
Solve arithmetic, geometric, Fibonacci, and polynomial sequences. Compute the nth term, partial series sums, convergence limits, and reverse-engineer unknown numerical patterns.
Term at Index a_n (n = 10)
19
Series Sum S_n
100
Progression Rate of Growth Curve
Rate curve plots indices up to first 15 values relative to peak heights. Negative values are color-coded red.
Step-by-step Mathematical Derivations
Discrete Progressions & Recurrence Foundations
Mathematical formulations for closed-form formulas, partial sums, and infinite convergence limits
In discrete mathematics, computer science, and numerical analysis, a sequence is an ordered enumeration of numbers governed by a deterministic rule. The two foundational linear and exponential progressions are:
Second-Order Sequences & Fibonacci Homogeneous Recurrence
When first differences are non-constant, polynomial sequences exhibit constant higher-order differences. A quadratic sequence follows aₙ = An² + Bn + C, where the constant second difference Δ² = 2A. In contrast, the classic Fibonacci sequence is a second-order linear recurrence relation:
Discrete Sequence Families & Convergence Behavior Matrix
Mathematical properties, explicit nth-term formulas, partial sums, and computational use cases
| Sequence Family | Recurrence Order | Explicit Formula (aₙ) | Series Partial Sum (Sₙ) | Infinite Convergence | Practical Applications |
|---|---|---|---|---|---|
| Arithmetic (Linear) | First Order (Δ¹ = d) | a₁ + (n - 1)d | (n/2)(a₁ + aₙ) | Diverges (±∞) | Linear depreciation, bond laddering, simple interest |
| Geometric (Exponential) | First Order Ratio (r) | a₁ × rⁿ⁻¹ | a₁(1 - rⁿ) / (1 - r) | S_∞ = a₁ / (1 - r) if |r| < 1 | Compound interest, half-life decay, bouncing mechanics |
| Quadratic (Polynomial) | Second Order (Δ² = 2A) | An² + Bn + C | ∑ An² + Bn + C | Diverges (+∞) | Kinematics, gravity acceleration, nested loops O(n²) |
| Triangular Numbers | Second Order | n(n + 1) / 2 | n(n + 1)(n + 2) / 6 | Diverges (+∞) | Complete graph edges, network handshakes, bowling pins |
| Fibonacci Recurrence | Second Order Linear | (φⁿ - ψⁿ) / √5 | Fₙ₊₂ - 1 | Diverges (+∞) | Phyllotaxis, Fibonacci heaps, algorithmic search |
| Harmonic Series | Reciprocal AP | 1 / [a₁ + (n-1)d] | Hₙ ~ ln(n) + γ | Diverges (Slowly) | Average speed calculations, load balancing, acoustics |
Frequently Asked Questions
Authoritative mathematical explanations on sequence rules, series sums, and pattern recognition
What is the mathematical difference between an arithmetic and geometric sequence?
An arithmetic sequence progresses by adding a constant common difference (d) to each term (a_n = a_1 + (n - 1)d), creating linear growth. A geometric sequence progresses by multiplying each preceding term by a constant common ratio (r) (a_n = a_1 × r^(n-1)), creating exponential growth or decay.
How does the calculator compute the partial sum of an arithmetic series?
The arithmetic series sum uses Carl Friedrich Gauss's pairing theorem: S_n = (n / 2) × [2a_1 + (n - 1)d] = n × (a_1 + a_n) / 2. This multiplies the average of the first and last terms by the total count of terms.
When does an infinite geometric series converge to a finite sum?
An infinite geometric series converges if and only if the absolute value of the common ratio is strictly less than one (|r| < 1). In this case, the infinite sum converges to S_∞ = a_1 / (1 - r). If |r| ≥ 1, the series diverges to infinity or oscillates indefinitely.
How does the auto-detect engine identify quadratic sequences?
The auto-detect engine computes first-order differences (Δ1 = a_(n+1) - a_n). If these are not constant, it computes second-order differences (Δ2 = Δ1_(n+1) - Δ1_n). If Δ2 is constant, the sequence is quadratic with general form a_n = An² + Bn + C, where 2A equals the second difference.
How does Binet's formula calculate the nth Fibonacci number directly?
Binet's closed-form formula computes any Fibonacci number without calculating preceding terms: F_n = (φ^n - ψ^n) / √5, where φ = (1 + √5) / 2 ≈ 1.6180339887 (the Golden Ratio) and ψ = (1 - √5) / 2 ≈ -0.6180339887.