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Fibonacci Numbers — Definition, Formula & Examples

Fibonacci numbers form a sequence where each number equals the sum of the two numbers before it, starting with 1, 1. The sequence begins 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and continues forever.

The Fibonacci sequence {Fn}\{F_n\} is defined recursively by F1=1F_1 = 1, F2=1F_2 = 1, and Fn=Fn−1+Fn−2F_n = F_{n-1} + F_{n-2} for all integers n≥3n \geq 3. Some authors use F0=0F_0 = 0 and F1=1F_1 = 1 as the starting values, which shifts the indexing but produces the same sequence of values.

Key Formula

Fn=Fn−1+Fn−2F_n = F_{n-1} + F_{n-2}
Where:
  • FnF_n = The nth Fibonacci number
  • Fn−1F_{n-1} = The Fibonacci number just before F_n
  • Fn−2F_{n-2} = The Fibonacci number two positions before F_n

How It Works

To build the Fibonacci sequence, start with 1 and 1, then keep adding the last two numbers to get the next one. For example, 1+1=21 + 1 = 2, then 1+2=31 + 2 = 3, then 2+3=52 + 3 = 5, and so on. This "add the previous two" rule is called a recursive rule because each term depends on earlier terms. Fibonacci numbers appear in surprising places — the number of petals on many flowers, the spiral pattern of seeds in a sunflower, and the branching of trees often follow Fibonacci patterns.

Worked Example

Problem: Find the 10th Fibonacci number.
Start: Write the first two Fibonacci numbers.
F1=1,F2=1F_1 = 1, \quad F_2 = 1
Build up: Apply the rule repeatedly: add the two most recent terms to get the next.
F3=1+1=2,F4=1+2=3,F5=2+3=5F_3 = 1+1 = 2, \quad F_4 = 1+2 = 3, \quad F_5 = 2+3 = 5
Continue: Keep going through positions 6 to 10.
F6=8,F7=13,F8=21,F9=34,F10=55F_6 = 8, \quad F_7 = 13, \quad F_8 = 21, \quad F_9 = 34, \quad F_{10} = 55
Answer: The 10th Fibonacci number is 55.

Another Example

Problem: Two consecutive Fibonacci numbers are 21 and 34. What is the next Fibonacci number after them?
Identify: The two most recent terms are 21 and 34.
Apply the rule: Add the two terms together.
21+34=5521 + 34 = 55
Answer: The next Fibonacci number is 55.

Visualization

Why It Matters

Fibonacci numbers appear in pre-algebra and algebra courses whenever students study patterns and recursive sequences. Computer science students use Fibonacci numbers to learn about recursion, algorithm efficiency, and dynamic programming. In nature, biologists observe Fibonacci counts in flower petals, pinecone spirals, and shell growth, making this sequence one of the most widely recognized bridges between mathematics and the real world.

Common Mistakes

Mistake: Starting the sequence with 1, 2 instead of 1, 1.
Correction: The standard Fibonacci sequence begins 1, 1 (or 0, 1). The number 2 is the third term, not the second.
Mistake: Confusing the Fibonacci rule with doubling or multiplying consecutive terms.
Correction: Each Fibonacci number is the sum (not the product) of the two before it. For example, F7=8+5=13F_7 = 8 + 5 = 13, not 8×5=408 \times 5 = 40.

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