Fibonacci Calculator Formula

Understand the math behind the fibonacci calculator. Each variable explained with a worked example.

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Formulas Used

Fib N

fib_n = round(pow((1 + sqrt(5)) / 2, n) / sqrt(5))

Golden Ratio

golden_ratio = (1 + sqrt(5)) / 2

Fib N Minus 1

fib_n_minus_1 = n >= 1 ? round(pow((1 + sqrt(5)) / 2, n - 1) / sqrt(5)) : 0

Ratio

ratio = n >= 2 ? round(pow((1 + sqrt(5)) / 2, n) / sqrt(5)) / round(pow((1 + sqrt(5)) / 2, n - 1) / sqrt(5)) : 0

Variables

VariableDescriptionDefault
nPosition (n)10

How It Works

Fibonacci Sequence

Definition

F(0) = 0, F(1) = 1, F(n) = F(n-1) + F(n-2)

Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

Binet's Formula

F(n) = (phi^n - psi^n) / sqrt(5)

where phi = (1 + sqrt(5))/2 (golden ratio) and psi = (1 - sqrt(5))/2.

For large n, psi^n approaches 0, so F(n) ≈ phi^n / sqrt(5).

Golden Ratio

As n grows, F(n)/F(n-1) approaches the golden ratio phi ≈ 1.6180339887.

Worked Example

Find the 10th Fibonacci number.

n = 10
  1. 01F(10) = phi^10 / √5
  2. 02= 1.61803^10 / 2.23607
  3. 03≈ 122.992 / 2.236
  4. 04≈ 55
  5. 05Sequence up to F(10): 0,1,1,2,3,5,8,13,21,34,55

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