Fibonacci Calculator
Free Fibonacci Calculator for sequences. Enter values to get step-by-step solutions with formulas and graphs. Free to use with no signup required.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Fibonacci Calculator
Calculator
Adjust values & calculateEnter your values below. Every result is computed in your browser — no data is sent to any server.
Formula: F(n) = F(n-1) + F(n-2)
Worked example — F(12) = 144
Formula
F(n) = F(n-1) + F(n-2)
Each Fibonacci number is the sum of the two immediately preceding numbers in the sequence, starting from F(1) = 1 and F(2) = 1. The ratio of consecutive terms converges to the golden ratio φ ≈ 1.618 as n grows.
Worked Examples
Example 1: Finding a specific term
Problem:What is the 12th Fibonacci number (using F(1)=1, F(2)=1)?
Solution:Building the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 — the 12th term is 144.
Result:F(12) = 144
Example 2: Approaching the golden ratio
Problem:Show how the ratio of consecutive Fibonacci terms approaches φ ≈ 1.618 as n grows.
Solution:F(10)/F(9) = 55/34 ≈ 1.6176. F(15)/F(14) = 610/377 ≈ 1.6180. The ratio converges toward φ with each additional term.
Result:Ratio converges to φ ≈ 1.6180339887
Frequently Asked Questions
What is the Fibonacci sequence?
The Fibonacci sequence is a series of numbers where each term is the sum of the two preceding ones, conventionally starting 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... It was introduced to Western mathematics by Leonardo of Pisa (known as Fibonacci) in his 1202 book Liber Abaci, though the pattern had already been described centuries earlier by Indian mathematicians studying poetic meter.
How is each Fibonacci number calculated?
Using the recurrence relation F(n) = F(n-1) + F(n-2), with F(1) = 1 and F(2) = 1 as the starting values (some sources begin the sequence with F(0) = 0). To find any term, simply add the previous two terms — F(9) = F(8) + F(7) = 21 + 13 = 34.
What is the connection between the Fibonacci sequence and the golden ratio?
As you move further into the Fibonacci sequence, the ratio of consecutive terms F(n)/F(n-1) converges toward the golden ratio, φ ≈ 1.6180339887. For example, 34/21 ≈ 1.6190 and 55/34 ≈ 1.6176 — both already very close to φ. This relationship is formalized by Binet's Formula, which expresses F(n) exactly in terms of φ.
Where does the Fibonacci sequence appear in nature?
Fibonacci numbers describe the spiral arrangement of seeds in a sunflower head, the number of petals on many flowers (lilies have 3, buttercups 5, some daisies 34 or 55), the branching patterns of trees, and the logarithmic spiral shape of a nautilus shell. This isn't mysticism — it emerges from efficient packing rules in plant growth (phyllotaxis), where each new leaf or seed grows at the golden angle (≈137.5°) from the last.
How is the Fibonacci sequence used in computer science and finance?
In computer science, Fibonacci numbers illustrate recursion, dynamic programming, and algorithmic complexity (naive recursive computation is exponential-time, while an iterative or memoized approach is linear-time). In finance, 'Fibonacci retracement' levels — 23.6%, 38.2%, 50%, 61.8% — are popular (though controversial) technical-analysis tools traders use to guess where a stock price might reverse after a trend.
What are Lucas numbers and how do they relate to Fibonacci numbers?
Lucas numbers follow the identical recurrence rule (each term is the sum of the two before it) but start with different seed values: 2, 1, 3, 4, 7, 11, 18, 29... They are closely related to Fibonacci numbers through several identities, such as L(n) = F(n-1) + F(n+1), and both sequences are special cases of the broader family of linear recurrence sequences.
How fast does the Fibonacci sequence grow?
Because each term is roughly φ ≈ 1.618 times the previous one, Fibonacci numbers grow exponentially — the 50th term already exceeds 12 billion, and the 100th term is a 21-digit number. This exponential growth is exactly why naive recursive implementations without memoization become impractically slow past roughly the 40th term.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
Related Calculators
🧮Fibonacci Closed Form Calculator
Calculate fibonacci closed form with inputs, formulas, and instant results.
🧮Annulus Area Calculator
Calculate annulus area with inputs, formulas, and instant results.
🧮Area Calculator
Calculate area with inputs, formulas, and instant results.
🧮Area of a Rectangle Calculator
Calculate the area, perimeter, and diagonal of a rectangle. Find missing sides from known area. Convert between metric and imperial area units.
🧮Area of Crescent Calculator
Calculate area of crescent with inputs, formulas, and instant results.
🧮Center of Mass Calculator
Calculate center of mass with inputs, formulas, and instant results.
🧮Centroid Calculator
Calculate centroid with inputs, formulas, and instant results.
🧮Chord Length Calculator
Calculate chord length with inputs, formulas, and instant results.