Fibonacci Sequence

Fibonacci Sequence

The sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . for which the next term is found by adding the previous two terms. This sequence is encountered in many settings, from population models to botany.

Note: The sequence of ratios of consecutive terms has the Golden Mean as its limit.

Fibonacci sequence: 1,1,2,3,5,8,13,21,34,55,89… with recursive formula: f₁=1, f₂=1, fₙ=fₙ₋₁+fₙ₋₂

Key Formula

F_n = F_{n-1} + F_{n-2}

Where:

$F_n$ = The nth Fibonacci number (the term you want to find)
$F_{n-1}$ = The term immediately before F_n
$F_{n-2}$ = The term two positions before F_n
$F_1 = 1,\; F_2 = 1$ = The starting values (seed values) of the sequence

Worked Example

Problem: Find the 10th term of the Fibonacci sequence.

Step 1: Write down the two starting values of the sequence.

F_1 = 1, \quad F_2 = 1

Step 2: Apply the rule F_n = F_{n-1} + F_{n-2} repeatedly to build up the sequence from the 3rd term onward.

F_3 = 1 + 1 = 2, \quad F_4 = 2 + 1 = 3, \quad F_5 = 3 + 2 = 5

Step 3: Continue generating terms.

F_6 = 5 + 3 = 8, \quad F_7 = 8 + 5 = 13, \quad F_8 = 13 + 8 = 21

Step 4: Compute the 9th and 10th terms.

F_9 = 21 + 13 = 34, \quad F_{10} = 34 + 21 = 55

Answer: The 10th Fibonacci number is 55.

Another Example

This example demonstrates the connection between the Fibonacci sequence and the Golden Mean, rather than simply computing a term. It shows a property of the sequence instead of just extending it.

Problem: Show that the ratio of consecutive Fibonacci numbers approaches the Golden Mean by computing F_n / F_{n-1} for n = 2 through n = 8.

Step 1: List the first 8 Fibonacci numbers.

1,\; 1,\; 2,\; 3,\; 5,\; 8,\; 13,\; 21

Step 2: Compute the ratio of each term to the one before it.

\frac{1}{1} = 1.000, \quad \frac{2}{1} = 2.000, \quad \frac{3}{2} = 1.500, \quad \frac{5}{3} \approx 1.667

Step 3: Continue for the remaining terms.

\frac{8}{5} = 1.600, \quad \frac{13}{8} = 1.625, \quad \frac{21}{13} \approx 1.615

Step 4: Notice the ratios oscillate around and converge toward the Golden Mean.

\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180

Answer: The ratios 1.000, 2.000, 1.500, 1.667, 1.600, 1.625, 1.615 are converging toward the Golden Mean φ ≈ 1.618.

Frequently Asked Questions

What is the difference between the Fibonacci sequence and a geometric sequence?

A geometric sequence multiplies each term by a fixed ratio to get the next term (e.g., 2, 6, 18, 54, … with ratio 3). The Fibonacci sequence adds the two previous terms to get the next. A geometric sequence has one seed value and a common ratio, while the Fibonacci sequence needs two seed values and has no fixed ratio between consecutive terms—though that ratio approaches the Golden Mean.

Does the Fibonacci sequence always start with 1, 1?

The classic Fibonacci sequence starts with F_1 = 1 and F_2 = 1. Some authors define F_0 = 0, making the sequence 0, 1, 1, 2, 3, 5, …, which is equally valid. The same recurrence rule F_n = F_{n-1} + F_{n-2} applies in both cases. Changing the starting values produces a different but related sequence (sometimes called a Lucas sequence or generalized Fibonacci sequence).

Where does the Fibonacci sequence appear in nature?

Fibonacci numbers show up in the number of petals on many flowers (lilies have 3, buttercups 5, daisies often 34 or 55). The spiral arrangement of seeds in a sunflower head and the scales of a pinecone also follow Fibonacci patterns. These patterns arise because Fibonacci-based growth produces the most efficient packing and distribution of biological structures.

Fibonacci Sequence vs. Arithmetic Sequence

	Fibonacci Sequence	Arithmetic Sequence
Definition	Each term is the sum of the two preceding terms	Each term is the previous term plus a fixed constant (common difference)
Formula	F_n = F_{n-1} + F_{n-2}	a_n = a_1 + (n − 1)d
Growth pattern	Approximately exponential (grows by a factor near φ ≈ 1.618 each step)	Linear (increases by the same amount each step)
Seed values needed	Two (F_1 and F_2)	One (a_1) plus the common difference d
Example	1, 1, 2, 3, 5, 8, 13, …	3, 7, 11, 15, 19, … (d = 4)

Why It Matters

The Fibonacci sequence is one of the most famous sequences in mathematics and appears in courses from pre-algebra through college-level discrete math. You will encounter it in problems about recursive formulas, proof by induction, and limits of sequences. Beyond the classroom, it models real phenomena such as population growth in biology and proportions in art and architecture.

Common Mistakes

Mistake: Adding the wrong two terms — for example, adding the current term to itself instead of adding the current term to the one before it.

Correction: Always add the two most recent terms. To find F_n, you need F_{n-1} and F_{n-2}, not F_{n-1} twice. Writing out the sequence in order helps you keep track.

Mistake: Confusing the term number (n) with the term value (F_n). For instance, thinking the 8th Fibonacci number is 8.

Correction: The position and the value are different. F_6 = 8 and F_8 = 21. If asked for the nth term, count carefully from the start of the sequence.

Related Terms

Sequence — General concept that includes Fibonacci as a special case
Term — Each individual number in the Fibonacci sequence
Golden Mean — Limit of the ratio of consecutive Fibonacci numbers
Limit — Describes how Fibonacci ratios converge to φ
Model — Fibonacci models population growth scenarios
Recursive Formula — The Fibonacci rule is a recursive definition
Arithmetic Sequence — Another common sequence type, uses addition of a constant
Geometric Sequence — Another common sequence type, uses multiplication by a constant