Find the nᵗʰ term, finite series sum, infinite series sum, and sequence expansion for arithmetic, geometric, and arbitrary-precision Fibonacci sequences.
Formula: aₙ = a₁ + (n - 1)d | Sₙ = n(a₁ + aₙ) / 2
Formula: aₙ = a₁ × r^(n - 1) | Sₙ = a₁(1 - rⁿ) / (1 - r)
Definition: F₀ = 0, F₁ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂ | Fast Doubling BigInt
In mathematics, a sequence is an ordered list of numbers where each member is called a term. A series is the sum of the terms of a sequence.
Each term is obtained by adding a fixed constant (the common difference d) to the previous term.
Each term is obtained by multiplying the previous term by a fixed non-zero factor (common ratio r).
Each term is the sum of the two preceding ones, starting from 0 and 1. Ratio of successive terms converges to the Golden Ratio (Φ ≈ 1.618).
An infinite geometric series converges to a finite sum if and only if the absolute value of the common ratio is strictly less than 1 (|r| < 1):
An explicit formula allows you to calculate any nᵗʰ term directly without knowing previous terms (e.g., aₙ = a₁ + (n - 1)d). A recursive formula defines each term in relation to the term preceding it (e.g., aₙ = aₙ₋₁ + d).
As n grows large, the ratio of successive Fibonacci numbers Fₙ / Fₙ₋₁ asymptotically approaches the Golden Ratio, denoted by the Greek letter Phi (Φ = (1 + √5) / 2 ≈ 1.6180339887...).
Instead of slow recursive calls or iterating one by one, the calculator uses the O(log n) Fast Doubling algorithm with native JavaScript BigInt, enabling calculations for n = 5,000 in less than 1 millisecond without precision loss.
When r = 1, every term in the sequence is identical to the first term a₁. The finite sum of the first n terms is simply Sₙ = n × a₁.