Fibonacci Sequence Calculator
Generates Fibonacci sequences with custom starting values and computes the nth term or cumulative sum using Binet's formula. Use it for math coursework, algorithm analysis, or exploring the golden ratio.
Last updated: September 2026
Formula below · 2 sources (oeis.org, Wikipedia) · Updated Sep 2026
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About this calculator
The Fibonacci sequence is defined by the recurrence F(n) = F(n−1) + F(n−2), with standard seeds F(0) = 0 and F(1) = 1. Binet's closed-form formula computes the nth term directly without iteration: F(n) = (φⁿ − ψⁿ) / √5, where φ = (1 + √5) / 2 ≈ 1.618 (the golden ratio) and ψ = (1 − √5) / 2 ≈ −0.618. The formula works because φ and ψ are roots of the characteristic equation x² = x + 1. The calculator builds the terms by the recurrence itself, so custom starting values (F₀ = a, F₁ = b) are exact: the nth term is a·F(n−1) + b·F(n). "Sum of First N Terms" adds F₀ through F(n−1), which with standard seeds equals F(n+1) − 1; "Generate Sequence" lists those n terms; "Golden Ratio Approximation" returns F(n)/F(n−1).
How to use
Find the 10th Fibonacci number using standard seeds F₀ = 0, F₁ = 1. Set n_terms = 10, start_a = 0, start_b = 1, calculation_type = 'nth_term'. The calculator applies the recurrence ten times and returns 55; Binet's formula agrees: F(10) = (φ¹⁰ − ψ¹⁰) / √5 = (122.9919 − 0.0081) / 2.2361 ≈ 55. Verify by listing: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 — the 10th term (0-indexed) is indeed 55. ✓
Frequently asked questions
What is Binet's formula and why does it give exact integers from irrational numbers?
Binet's formula, F(n) = (φⁿ − ψⁿ)/√5, expresses Fibonacci numbers in terms of the irrational golden ratio φ = (1+√5)/2 and its conjugate ψ = (1−√5)/2. Despite involving irrationals, the result is always an integer because φ and ψ are roots of the same integer-coefficient polynomial, so their powers combine to cancel all irrational parts. In practice, |ψ| < 1 means ψⁿ → 0 rapidly, so F(n) is simply the nearest integer to φⁿ/√5. For very large n, floating-point rounding errors accumulate, so iterative methods are preferred for precision beyond roughly n = 70.
How does the ratio of consecutive Fibonacci numbers relate to the golden ratio?
As n increases, the ratio F(n+1)/F(n) converges to φ = (1+√5)/2 ≈ 1.6180339…. For small n the ratio oscillates — F(2)/F(1) = 1, F(3)/F(2) = 2, F(5)/F(4) = 1.6667 — but it settles quickly: F(13)/F(12) = 233/144 ≈ 1.6181. This connection to φ appears in nature (spiral phyllotaxis in sunflowers and pinecones), in Renaissance art and architecture, and in efficient search algorithms. The golden ratio is also the positive solution to the equation x² = x + 1, which is the characteristic equation of the Fibonacci recurrence.
What is the sum of the first n Fibonacci numbers and how is it calculated?
The sum of the first n Fibonacci numbers (F(1) through F(n), 1-indexed) equals F(n+2) − 1. This elegant identity — sometimes called the Fibonacci identity for sums — can be proved by induction or by telescoping the recurrence. For example, summing F(1) through F(7): 1+1+2+3+5+8+13 = 33, and F(9) − 1 = 34 − 1 = 33. ✓ In this calculator "Sum of First N Terms" starts at F(0) = 0, so N terms (F(0) to F(N−1)) sum to F(N+1) − 1; to get F(1) through F(7) = 33, enter N = 8.