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

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Cory Mitchell
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Cory Mitchell, CMT is the founder of TradeThatSwing.com. He has been a professional day and swing trader since 2005. Cory is an expert on stock, forex and futures price action trading strategies.
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Updated September 30, 2023
Reviewed by Charles Potters
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Jared Ecker
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Fact checked by Jared Ecker
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Jared Ecker is a researcher and fact-checker. He possesses over a decade of experience in the Nuclear and National Defense sectors resolving issues on platforms as varied as stealth bombers to UAVs. He holds an A.A.S. in Aviation Maintenance Technology, a B.A. in History, and a M.S. in Environmental Policy & Management.
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Fibonacci Sequence

Investopedia / Yurle Villegas

What Is the Fibonacci Sequence?

The Fibonacci sequence was developed by the Italian mathematician, Leonardo Fibonacci, in the 13th century. The sequence of numbers, starting with zero and one, is a steadily increasing series where each number is equal to the sum of the preceding two numbers.

Some traders believe that the Fibonacci numbers and ratios created by the sequence play an important role in finance that traders can apply using technical analysis.

Key Takeaways

  • The Fibonacci sequence is a set of steadily increasing numbers where each number is equal to the sum of the preceding two numbers.
  • The golden ratio of 1.618 is derived from the Fibonacci sequence.
  • Many things in nature have dimensional properties that adhere to the golden ratio of 1.618.
  • The Fibonacci sequence can be applied to finance by using four techniques including retracements, arcs, fans, and time zones.

Understanding the Fibonacci Sequence

The numbers in the Fibonacci Sequence don't equate to a specific formula, however, the numbers tend to have certain relationships with each other. Each number is equal to the sum of the preceding two numbers. For example, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377.

Fibonacci Sequence Rule

xn = xn−1 + xn−2

where:

xn is term number "n"

xn−1 is the previous term (n−1)

xn−2 is the term before that (n−2)

The golden ratio of 1.618, important to mathematicians, scientists, and naturalists for centuries is derived from the Fibonacci sequence. The quotient between each successive pair of Fibonacci numbers in the sequence approximates 1.618, or its inverse 0.618.

Golden Ratio

The golden ratio is derived by dividing each number of the Fibonacci series by its immediate predecessor. Where F(n) is the nth Fibonacci number, the quotient F(n)/ F(n-1) will approach the limit 1.618, known as the golden ratio.

Many things in nature have dimensional properties that adhere to the ratio of 1.618, like the honeybee. If you divide the female bees by the male bees in any given hive, you will get a number near 1.618. The golden ratio also appears in the arts and rectangles whose dimensions are based on the golden ratio appear at the Parthenon in Athens and the Great Pyramid in Giza.

How to Use the Fibonacci Sequence

The Fibonacci sequence can be applied to finance by using four techniques including retracements, arcs, fans, and time zones.

Fibonacci retracements require two price points chosen on a chart, usually a swing high and a swing low. Once two points are chosen, the Fibonacci numbers and lines are drawn at percentages of that move. If a stock rises from $15 to $20, then the 23.6% level is $18.82, or $20 - ($5 x 0.236) = $18.82. The 50% level is $17.50, or $15 - ($5 x 0.5) = $17.50.

Image

Image by Sabrina Jiang © Investopedia 2021

Fibonacci retracements are the most common form of technical analysis based on the Fibonacci sequence. During a trend, Fibonacci retracements can be used to determine how deep a pullback may be. Traders tend to watch the Fibonacci ratios between 23.6% and 78.6% during these times. If the price stalls near one of the Fibonacci levels and then start to move back in the trending direction, an investor may trade in the trending direction.

Arcs, fans, and time zones are similar concepts but are applied to charts in different ways. Each one shows potential areas of support or resistance, based on Fibonacci numbers applied to prior price moves. These supportive or resistance levels can be used to forecast where prices may fall or rise in the future.

What Is the Fibonacci Spiral?

The limits of the squares of successive Fibonacci numbers create a spiral known as the Fibonacci spiral. It follows turns by a constant angle close to the golden ratio and is commonly called the golden spiral. The numbers of spirals in pinecones are Fibonacci numbers, as is the number of petals in each layer of certain flowers. In spiral-shaped plants, each leaf grows at an angle compared to its predecessor, and sunflower seeds are packed in a spiral formation in the center of their flower in a geometry governed by the golden ratio.

Where Is the Fibonacci Sequence Evident?

In almost all flowering plants, the number of petals on the flower is a Fibonacci number. It is extremely rare for the number of petals not to be so and examples of this phenomenon include corn marigold, cineraria, and daisies with 13 petals and asters and chicory with 21 petals.

How Can the Fibonacci Sequence Affect Trading Behavior?

Humans tend to identify patterns and traders easily equate patterns in charts through the Fibonacci sequence. It's unproven that Fibonacci numbers relate to fundamental market forces, however, markets by design react to the beliefs of their players. Consequently, if investors buy or sell because of Fibonacci analysis, they tend to create a self-fulfilling prophecy that affects the market trends.

The Bottom Line

The Fibonacci sequence is a set of steadily increasing numbers where each number is equal to the sum of the preceding two numbers. Many things in nature have dimensional properties that adhere to the golden ratio of 1.618, a quotient derived from the Fibonacci sequence. When applied to finance and trading, investors apply the Fibonacci sequence through four techniques including retracements, arcs, fans, and time zones.

Article Sources
Investopedia requires writers to use primary sources to support their work. These include white papers, government data, original reporting, and interviews with industry experts. We also reference original research from other reputable publishers where appropriate. You can learn more about the standards we follow in producing accurate, unbiased content in our editorial policy.

  1. Wild Maths. "Fibonacci and Bees."

  2. Smithsonian Magazine. "The Fibonacci Sequence Is Everywhere - Even the Troubled Stock Market."

  3. Science Struck. "13 Real-Life Examples of the Golden Ratio."

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