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 by Manoj Kumar, Mathematics Educator
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.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy