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# Fibonacci Numbers — Golden Ratio and Binet's Formula

[Algebra](/content/tag/algebra/index.html)

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.

BT  
[Bhanzu Team](/content/authors/bhanzu-team/index.html) 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](/content/math/algebra/sequences/index.html): 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:

- **Sum of the first n terms:** F_0 + F_1 + ⋯ + F_n = F_{n+2} - 1.
- **Every third number is even:** 0, 2, 8, 34, 144,…  — the even terms sit at F_0, F_3, F_6,….
- **Cassini's identity:** \( F_{n+1} F_{n-1} - F_n^2 = (-1)^n \).
- **Consecutive terms are coprime:** \( \gcd(F_n, F_{n+1}) = 1 \).

## 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](https://mathshistory.st-andrews.ac.uk/Biographies/Fibonacci/) (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:

- **Plant growth.** Sunflower seeds and pinecone scales spiral counts that are almost always consecutive Fibonacci numbers.
- **Computer science.** The Fibonacci search technique and Fibonacci heaps
- **Finance.** Traders use