Fibonacci Numbers — Golden Ratio and Binet's Formula

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Fibonacci Numbers — Golden Ratio and Binet's Formula

Algebra

TL;DR

The Fibonacci numbers are the sequence 0,1,1,2,3,5,8,13,…, where each term is the sum of the two before it: F_n = F_{n-1} + F_{n-2}. This article covers the recurrence, the full list, the golden-ratio connection, Binet's formula for the n-th term, key properties, and six worked examples.

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Bhanzu Team Last updated on June 10, 2026 10 min read

What Are the Fibonacci Numbers?

The Fibonacci numbers are a sequence in which each number is the sum of the two numbers immediately before it, starting from 0 and 1. The sequence runs:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,…

The defining relationship is a recurrence — a rule that defines each term using earlier terms:

F_n = F_{n-1} + F_{n-2}, F_0 = 0, F_1 = 1.

So F_2 = F_1 + F_0 = 1, F_3 = F_2 + F_1 = 2, and so on. This makes Fibonacci a special kind of sequence in algebra: not arithmetic (no constant difference) and not geometric (no constant ratio), but recursive — each step depends on the two steps before.

What Is the Fibonacci Numbers List?

Here are the first fifteen Fibonacci numbers, indexed from F_0. Knowing the start of the list by sight makes most problems faster.

n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
F_n 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377

Does the sequence start at 0 or 1? The modern convention starts at F_0 = 0. Some older texts start the list at 1,1,2,3,… and call the first 1 either F_1 or F_0. The numbers are the same; only the index label shifts. This article uses F_0 = 0 throughout.

How Are the Fibonacci Numbers Related to the Golden Ratio?

Divide any Fibonacci number by the one before it, and the answer creeps toward a single fixed value. As n grows, ( \frac{F_{n+1}}{F_n} ) approaches the golden ratio:

( \varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887.)

Watch it converge: 85=1.6, 138=1.625, 2113≈1.615, 3421≈1.619. Each ratio lands closer to ( \varphi ). The golden ratio is the only positive number satisfying ( \varphi^2 = \varphi + 1 ) — which is just the Fibonacci rule written for a single number instead of a list.

What Is Binet's Formula?

Binet's formula computes the n-th Fibonacci number directly — no need to build up every earlier term:

( F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}, \quad \varphi = \frac{1+\sqrt{5}}{2}, \psi = \frac{1-\sqrt{5}}{2}.)

Here ( \psi \approx -0.618 ) is the "conjugate" of ( \varphi ). For reasonable n, you can compute ( \frac{\varphi^n}{\sqrt{5}} ) and round to the nearest whole number. Two surprises live in this formula: an integer sequence comes out of irrationals, and a "closed form" exists at all for something defined recursively. Example 4 below runs the calculation.

What Are the Properties of the Fibonacci Numbers?

A short list of patterns:

Examples of Fibonacci Numbers

Example 1

Find the next three Fibonacci numbers after 21, 34, 55.

Add the last two each time:

34 + 55 = 89, 55 + 89 = 144, 89 + 144 = 233.

Final answer: 89, 144, 233.

Example 2

Find the 8th Fibonacci number, F_8.

Counting from F_0:

F_8 = 21.

Final answer: F_8 = 21.

Example 3

A common slip — find F_6.

Correct: F_6 = 8.

Final answer: F_6 = 8.

Example 4

Use Binet's formula to find F_7.

Final answer: F_7 = 13.

Example 5

Estimate F_{15} using the golden ratio.

Final answer: F_{15} = 610.

Example 6

Verify the sum property: add F_0 through F_6.

Final answer: both sides equal 20.

The Mathematician Behind the Fibonacci Numbers

Leonardo of Pisa (c. 1170–1250, Italy), known as Fibonacci, introduced the sequence to Europe in his 1202 book Liber Abaci.

Why the Fibonacci Numbers Matter

The reason Fibonacci numbers fascinate isn't just arithmetic — it's that nature appears to use them: