# Fibonacci Sequence: Definition, How It Works, and How to Use It

![Fibonacci Sequence](https://www.investopedia.com/thmb/S4eG3EIC_Y04wq9l7Vr0bgcV4mY=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/FIBONACCI-SEQUENCE-ILLUSTRATION-Final-blue-2b2f4e8866c0420daf28748e90e4e765.jpg)

*— Investopedia / Yurle Villegas*

> **Definition**
>
> The Fibonacci sequence is a series of numbers where each successive number is equal to the sum of the two numbers that precede it.

The Fibonacci sequence is one of mathematics' most intriguing patterns, influencing fields ranging from nature and art to the financial markets. This numerical sequence, which begins with 0, 1, and continues by adding the previous two numbers, has been investigated for centuries.

What makes this sequence particularly fascinating is its ubiquitous presence in seemingly unrelated phenomena, from the spiral patterns of galaxies to the breeding patterns of animals.

In financial markets, traders and analysts have used Fibonacci relationships to identify potential market turning points and interest price levels. The idea is that markets, like many natural and social systems, tend to move in cycles and waves that reverse or pause at mathematically proportional levels, just like how a plant's growth might follow precise mathematical ratios.

### Key Takeaways

*   The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones.
*   It is closely related to the golden ratio, which appears in various natural and artistic contexts.
*   The sequence is used in financial markets to predict price movements through techniques like Fibonacci retracements, extensions, fans, and channels.
*   The Fibonacci sequence is evident in natural patterns, such as the arrangement of leaves and flowers.

## What Is the Fibonacci Sequence?

The Fibonacci sequence, commonly attributed to medieval Italian mathematician Leonardo Fibonacci (c. 1170-1250), has a rich history spanning multiple civilizations and millennia. Fibonacci, also known as Leonardo of Pisa, formally introduced the sequence to Western mathematics in his 1202 book _Liber Abaci_ (_Book of Calculation_). The sequence begins with 0 and 1, with each subsequent number being the sum of the previous two according to the formula xn = (xn−1) + (xn−2):

> 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 ...

The power of the Fibonacci sequence lies in its fundamental nature as a growth pattern. Each number is the sum of all previous growth plus the current growth, creating an organic expansion that mirrors many natural and artificial phenomena.

The Fibonacci sequence demonstrates several key properties:

1.  The ratio among consecutive numbers converges to about 1.618 (the "[golden ratio](https://www.investopedia.com/articles/technical/04/033104.asp)"), creating a natural scaling factor.
2.  The sequence exhibits self-similarity, meaning patterns repeat or resemble themselves at different scales.
3.  Each number can be expressed as the sum of nonconsecutive previous numbers, suggesting multiple paths to the same outcome.
4.  The sequence shows how simple rules can create complex but predictable patterns.

> **Tip**
>
> Every third number in the sequence is even, and the sum of any 10 consecutive Fibonacci numbers is divisible by 11.

## A Brief History of the Fibonacci Sequence

Fibonacci initially discovered this sequence while studying rabbit population growth under ideal conditions. The problem posed was, if we start with a pair of rabbits, how many pairs would there be after a year if each pair produces a new pair every month and new pairs become productive after two months? This seemingly simple question led to one of mathematics' most influential sequences.

However, the sequence's true origins trace back much further. Sanskrit scholars had described similar patterns as early as 200 BCE, with Indian mathematician Pingala using them in his work on patterns and rhythms. By 450 CE, another Indian mathematician, Virahanka, had explicitly described the pattern in his work on Sanskrit meters. The sequence later appeared in Hemachandra's work (about 1150 CE), predating Fibonacci's work by half a century.

In 1611, Johannes Kepler commented on the relationship between the Fibonacci sequence and the golden ratio, though he didn't pursue this connection extensively. It wasn't until the 19th century that the sequence was studied systematically by mathematicians such as François Édouard Anatole Lucas (1842-1891), who developed related sequences and explored their properties.

The sequence's application to financial markets emerged in the 1930s, when Ralph Nelson Elliott developed his [Elliott wave theory](https://www.investopedia.com/terms/e/elliottwavetheory.asp), incorporating Fibonacci relationships into market analysis. In the 1940s, technical analyst Charles Collins first explicitly used Fibonacci ratios to predict market moves.

The concept gained wider popularity in the 1970s through the work of [market technicians](https://www.investopedia.com/articles/professionaleducation/10/market-technicians-association.asp) like [W.D. Gan,n](https://www.investopedia.com/articles/investing/091615/mysterious-life-trading-legend-wd-gann.asp) who observed that market movements often corresponded with these mathematical relationships. This led to modern Fibonacci trading tools such as retracements, extensions, fans, and time studies.

### The Golden Ratio

Hidden in the Fibonacci sequence is the "divine proportion," or "golden ratio." Dividing two consecutive Fibonacci numbers converges to about 1.618.

This number appears throughout nature and design: galaxy spirals, nautilus shells, and tree branch arrangements. Artists and architects have long employed the golden ratio, which is evident in works like the Great Pyramid in Giza, the Mona Lisa, and the Parthenon, as well as modern structures like the U.N. Secretariat Building and the Sydney Opera House. Even the standard dimensions for credit cards were chosen to reflect its proportions.

Most relevantly, the ratio is said to emerge in financial markets, where price reversals may often occur at levels predicted by it, hinting at its subtle influence on human behavior.

## Using the Fibonacci Sequence in Financial Markets

The application of Fibonacci numbers in financial markets might seem like an unlikely leap from the Mona Lisa, rabbits, and sunflowers-but these mathematical relationships have now become common tools in [technical analysis](https://www.investopedia.com/terms/t/technicalanalysis.asp).

Financial markets often mirror natural patterns because markets are, essentially, collective human behavior actualized through price discovery, supply, and demand. Just as tree branches grow in Fibonacci proportions to optimize sunlight exposure, price movements may retrace and extend in these proportions because they represent optimal equilibrium points between buyers and sellers.

Traders don't typically use the sequence itself (0, 1, 1, 2, 3, 5, 8...) but [key ratios and proportions](https://www.investopedia.com/articles/trading/05/advfibonacci.asp) that derive from it, particularly 23.6%, 38.2%, 61.8%, and 100%.

[These levels](https://www.investopedia.com/articles/active-trading/091615/how-set-fibonacci-retracement-levels.asp) are used in the following circumstances by traders:

*   Major [support or resistance](https://www.investopedia.com/trading/support-and-resistance-basics/) levels
*   Trend lines or long-term moving averages
*   Previous significant price points
*   Psychologically important round numbers

> **Important**
>
> Much of the success of Fibonacci numbers also has to do with market psychology: When enough traders and investors begin to watch the same levels, those levels become self-fulfilling.

### Fibonacci Retracements

[Retracements](https://www.investopedia.com/terms/f/fibonacciretracement.asp) act like rubber bands: When prices move significantly in either direction, they often "snap back" via a [correction](https://www.investopedia.com/terms/t/technical_correction.asp) before continuing. These retracements may stop at predictable levels:

*   23.6% - Common in strong trends
*   38.2% - Often seen in bull markets
*   50.0% - Not a Fibonacci number, but psychologically important
*   61.8% - The golden ratio retracement
*   78.6% - Usually signals trend weakness

**Example:**  
If a stock rises from $100 to $200:

*   38.2% retracement = $161.80 (potential buying point)
*   61.8% retracement = $138.20 (major support level)

![Fibonacci retracement](https://www.investopedia.com/thmb/7PDy6owdUuo7Zz259Q3VSW3Vyzs=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/FibonacciRetracement-5c549bc5c9e77c0001cff8b0.png)

*TradingView.*

### Fibonacci Extensions

While retracements help find pullback entries, [extensions](https://www.investopedia.com/terms/f/fibonacciextensions.asp) suggest potential profit targets:

*   127.2% - First major extension
*   161.8% - Golden ratio target
*   261.8% - Major long-term target

**Example:**  
After our $100 to $200 move and successful retracement, we have two long-term potential profit-taking points:

*   161.8% target = $261.80
*   261.8% target = $361.80  
    

![Fibonacci extension](https://www.investopedia.com/thmb/BYrkrHMtfE01YVxLNazU-GZaL5U=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/FibonacciExtensions-5c549cfb46e0fb00018200cc.png)

*TradingView.*

> **Tip**
>
> A Fibonacci cluster occurs when several repeated Fibonacci retracements or extensions converge around a specified price range.

### Fibonacci "Time Zones"

[Time-based tools](https://www.investopedia.com/terms/f/fibonaccitimezones.asp) help anticipate when price moves might occur. Vertical lines spaced according to Fibonacci numbers help traders identify potential timing for market reversals or continuations. Each line is spaced a Fibonacci number of periods apart (1, 2, 3, 5, 8, 13, etc.), which is useful for any time frame (minutes, hours, days, weeks, etc.).  

![Fibonacci Time Zones](https://www.investopedia.com/thmb/GXMsk4jhB0g2UqJwVdWGMeY60EA=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/FibonacciTimesZones-5c86af7546e0fb0001cbf566.png)

*TradingView.*

### Fibonacci Fans

[Fans](https://www.investopedia.com/terms/f/fibonaccifan.asp) are diagonal lines drawn using Fibonacci ratios to identify potential support and resistance levels as price moves across time. The lines are drawn at angles determined by 38.2%, 50%, and 61.8% levels.

![Brent Crude Oil Fibonacci Fan](https://www.investopedia.com/thmb/jBUngnMQ2YFcQFqIqunYIh7CLBI=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/UKOIL_FibonacciFan-0bbb90fc6e454e0f8c629790d72e736f.png)

*Brent Crude Oil Fibonacci Fan. — Tradingview*

### Fibonacci Arcs

[Fibonacci arcs](https://www.investopedia.com/terms/f/fibonacciarc.asp) are created by drawing circular arcs intersecting a trend line at key Fibonacci ratios, these arcs can identify potential support and resistance levels based on both price and time. The arcs are drawn, once again, at 38.2%, 50%, and 61.8% of the price range.

![US 10 Year Yield Fibonacci Arc](https://www.investopedia.com/thmb/-pepTlC8TuyuI7ihEIc3Z-lpbFE=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/US10Y_FibonacciArc-a6a8ec3b983b4292988ff7315c5c60e7.png)

*US 10 Year Yield Fibonacci Arc. — Tradingview*

## The Bottom Line

The Fibonacci sequence is one of mathematics' most versatile and widely applicable concepts. In financial markets, traders have adapted these mathematical relationships as practical tools for market analysis.

Technical traders use ratios and levels derived from the Fibonacci sequence to help identify support and resistance, as well as trends and reversals, with tools ranging from retracements and extensions to fans and arcs.

> **Article Sources**
>
> 1.  Omotehinwa, T. O., & Ramon, S. O. (2013). "[Fibonacci Numbers and Golden Ratio in Mathematics and Science](https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=a6429311640b2ae555e6f9c54f615d215090438f)." _International Journal of Computer and Information Technology_, vol. 2, no. 4, pp. 630-638.
>     
> 2.  Math! Science! History! "[Fibonacci and his Rabbits](https://mathsciencehistory.com/2025/01/28/fibonacci-and-his-rabbits/)."
>     
> 3.  Purram, V., Rao, G. S., & Charanarur, P. "[The Indian Mathematical Philosophy and Its Overlooked Connections: Fibonacci Numbers and Their Fractal Properties](https://www.researchgate.net/profile/Vaishnnavi-Purram/publication/387603671_The_Indian_Mathematical_Philosophy_and_its_Overlooked_Connections_Fibonacci_Numbers_and_their_Fractal_Properties/links/6775718f00aa3770e0ccead4/The-Indian-Mathematical-Philosophy-and-its-Overlooked-Connections-Fibonacci-Numbers-and-their-Fractal-Properties.pdf)."
>     
> 4.  ETH Zurich. "[Johannes Kepler – the golden ratio in "World Harmonic.](https://library.ethz.ch/en/locations-and-media/platforms/virtual-exhibitions/fibonacci-un-ponte-sul-mediterraneo/reception-of-fibonacci-numbers-and-the-golden-ratio/johannes-kepler-the-golden-ratio-in-world-harmonic.html)"
>     
> 5.  University of Evansville. "[Édouard Lucas (1842-1891) Number Theorist](https://faculty.evansville.edu/ck6/bstud/lucas.html)."
>     
> 6.  Lo, A. W., & Hasanhodzic, J. (2010). [_The evolution of technical analysis: Financial prediction from Babylonian tablets to Bloomberg terminals_](https://books.google.co.il/books?hl=en&lr=&id=aJv9DwAAQBAJ&oi=fnd&pg=PR7#v=onepage&q&f=false) (Vol. 96). John Wiley & Sons.
>     
> 7.  The Mona Lisa Foundation. "[Leonardo and Mathematics](https://monalisa.org/2012/09/12/leonardo-and-mathematics-in-his-paintings/)."
>     
> 8.  The Golden Number. "[The UN Secretariat Building, Le Corbusier and the Golden Ratio](https://www.goldennumber.net/un-secretariat-building-golden-ratio-architecture/)."
>     
> 9.  The Golden Number. "[Credit Cards and Golden Ratio Proportions](https://www.goldennumber.net/credit-cards/#:~:text=Standard%20sized%20credit%20cards%20are,0.618%2C%20the%20reciprocal%20of%201.618.)."
>     
> 10.  Fischer, R., & Fischer, J. (2001). [_The New Fibonacci Trader: Tools and Strategies for Trading Success_](https://books.google.co.il/books?hl=en&lr=&id=k-gHJHZ6nBsC&oi=fnd&pg=PA36#v=onepage&q&f=false). John Wiley & Sons.
>      
> 11.  Khan, K. M., Rehman, W., & Saif, O. B. (2022). [Predicting key reversal points through Fibonacci retracements](https://readersinsight.net/jmi/article/view/2638). _Journal of Management Info_, _9_(3), 299-310.