Fibonacci Sequence

Learn the Fibonacci sequence, where each term is the sum of the two before it, its hidden patterns, and how the ratio of neighbours approaches the golden ratio.

Intermediate25 min18 problems
Practice Problems

Definition

The Fibonacci sequence is the list of numbers
1,1,2,3,5,8,13,21,34,55,89,144,…1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \ldots
in which every term after the first two is the sum of the two terms before it: 1+1=21 + 1 = 2, 1+2=31 + 2 = 3, 2+3=52 + 3 = 5, 3+5=83 + 5 = 8, and so on.

The Recursive Rule

We call the terms F1,F2,F3,…F_1, F_2, F_3, \ldots and write:
F1=1,F2=1,Fn=Fn−1+Fn−2 for n≥3F_1 = 1, \quad F_2 = 1, \quad F_n = F_{n-1} + F_{n-2} \text{ for } n \geq 3
You will also see it written as F(n)=F(n−1)+F(n−2)F(n) = F(n-1) + F(n-2). This is a recursive rule: to get a term you need the terms just before it. In an arithmetic sequence you always add the same dd, in a geometric sequence you always multiply by the same rr, but here the amount you add keeps growing.

Fibonacci's Rabbits

In 1202, Leonardo of Pisa, known as Fibonacci, put a rabbit puzzle in his book *Liber Abaci*. Today it is usually told like this: one newborn pair of rabbits is placed in a field. A pair is grown up after one month, and from then on every grown-up pair has one new pair each month. No rabbit dies. Counting the pairs month by month gives 1,1,2,3,5,8,…1, 1, 2, 3, 5, 8, \ldots (Fibonacci's own pair had babies from the first month, so his count began 1,2,3,5,…1, 2, 3, 5, \ldots, the same numbers. Scholars in India had already found them centuries earlier while counting rhythms of long and short syllables in poetry.)

Patterns Hidden Inside

  • Every third term is even: F3=2F_3 = 2, F6=8F_6 = 8, F9=34F_9 = 34, F12=144F_{12} = 144, and all other terms are odd.
  • **Sum of the first nn terms**: F1+F2+⋯+Fn=Fn+2−1F_1 + F_2 + \cdots + F_n = F_{n+2} - 1. For example 1+1+2+3+5=12=13−11 + 1 + 2 + 3 + 5 = 12 = 13 - 1.
  • Ratio of neighbours: Fn+1Fn\frac{F_{n+1}}{F_n} gets closer and closer to the golden ratio
φ=1+52≈1.618\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.618

Try it now

Which number comes next? 1,1,2,3,5,8,13,…1, 1, 2, 3, 5, 8, 13, \ldots

Worked Examples

The Fibonacci sequence starts 1,1,2,3,5,8,13,21,34,…1, 1, 2, 3, 5, 8, 13, 21, 34, \ldots Find F10F_{10}, F11F_{11} and F12F_{12}.
1

Find the last two known terms

F8=21F_8 = 21 and F9=34F_9 = 34 → F8=21F_8 = 21, F9=34F_9 = 34
2

Add them to get F10F_{10}

F10=F9+F8=34+21F_{10} = F_9 + F_8 = 34 + 21 → F10=55F_{10} = 55
3

Add the two newest terms to get F11F_{11}

F11=F10+F9=55+34F_{11} = F_{10} + F_9 = 55 + 34 → F11=89F_{11} = 89
4

Repeat once more for F12F_{12}

F12=F11+F10=89+55F_{12} = F_{11} + F_{10} = 89 + 55 → F12=144F_{12} = 144

Common Mistakes

Doubling the last term: 1,1,2,4,8,16,…1, 1, 2, 4, 8, 16, \ldots

Why it's wrong: Doubling gives a geometric sequence. The Fibonacci rule adds the last TWO terms, and they are not equal (except at the very start).

Correct: After 22 comes 1+2=31 + 2 = 3, then 2+3=52 + 3 = 5: 1,1,2,3,5,8,…1, 1, 2, 3, 5, 8, \ldots

Miscounting positions, for example saying F10=89F_{10} = 89

Why it's wrong: The first two terms are both 11, so it is easy to skip one. Some books also put a 00 first (0,1,1,2,…0, 1, 1, 2, \ldots); if you count that 00 as the first term, every position shifts by one.

Correct: Write positions above the terms: F1=1F_1 = 1, F2=1F_2 = 1, F3=2F_3 = 2, ..., F10=55F_{10} = 55, F11=89F_{11} = 89. Check which start a book uses.

Thinking the ratios climb steadily up to 1.6181.618, or that one ratio is exactly φ\varphi

Why it's wrong: The ratios jump above and below φ\varphi: 1.61.6, 1.6251.625, 1.6151.615, 1.6191.619, ... Every ratio is a fraction, and φ\varphi is irrational, so no ratio ever equals it.

Correct: Say the ratios approach φ\varphi from both sides and get as close as we like, but never reach it.

Believing every spiral and every beautiful rectangle is Fibonacci or golden

Why it's wrong: A nautilus shell is a spiral, but its growth ratio is not φ\varphi. Many golden ratio claims about the Parthenon or the Mona Lisa depend on where you choose to draw the lines.

Correct: Count or measure first. Sunflower and pine cone spiral counts are often Fibonacci numbers, but not always, and art claims need real measurements.

Interactive Visual

Add squares of side 1, 1, 2, 3, 5, 8... and watch the ratio approach 1.618

11235

Rectangle: 5 × 8

The sequence

11235

2 + 3 = 5

Each new term is the sum of the two terms before it.

Ratio of neighbors: new term ÷ term before

  • 1 ÷ 1 = 1< φ
  • 2 ÷ 1 = 2> φ
  • 3 ÷ 2 = 1.5< φ
  • 5 ÷ 3 = 1.66667> φ
1φ2step

The ratios jump above and below φ = 1.618034 and get closer every time (now off by 0.048633).

Fibonacci's rabbits: pairs month by month

A baby pair grows up in one month. Every adult pair has a baby pair each month. No rabbit ever leaves.

MonthPairsAdultBabyTotal
1011
2101
3112
4213

adult pairbaby pair

Pairs of rabbits, month by month

1123

1 + 2 = 3

Each new term is the sum of the two terms before it.

Ratio of neighbors: new term ÷ term before

  • 1 ÷ 1 = 1< φ
  • 2 ÷ 1 = 2> φ
  • 3 ÷ 2 = 1.5< φ
1φ2step

The ratios jump above and below φ = 1.618034 and get closer every time (now off by 0.118034).

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Practice Problems

18 problems
Problem 1 of 18
Easy
Which number comes next? 1,1,2,3,5,8,13,…1, 1, 2, 3, 5, 8, 13, \ldots

Why It Matters

The Fibonacci sequence is one of the best known patterns in mathematics, and it shows up in surprising places:
  • Plants: many sunflower heads show 34 spirals one way and 55 the other, pine cones often show 8 and 13, and pineapples 8 and 13 as well. Packing seeds at a constant turning angle (about 137.5°137.5°, linked to φ\varphi) packs them evenly with hardly any gaps, and Fibonacci spiral counts follow from that.
  • Counting: the number of ways to climb a staircase taking 1 or 2 steps at a time is a Fibonacci number.
  • Units: because φ≈1.618\varphi \approx 1.618 is close to 1.6091.609 (kilometers in a mile), neighbouring Fibonacci numbers convert miles to kilometers: 5 miles is about 8 km, 8 miles about 13 km.
  • Recursive thinking: building each step from the steps before is how many computer programs and growth models work.
A word of honesty: not every spiral in nature is a Fibonacci spiral, plant counts sometimes miss by one, and many claims about the golden ratio in famous paintings and buildings are exaggerated. Good mathematicians check the numbers before believing the story.

Real World Applications

Sunflower Spirals

The seeds in a sunflower head form two families of spirals, one turning clockwise and one counterclockwise. Their numbers are very often two neighbouring Fibonacci numbers.

Example:

A typical sunflower shows 34 spirals one way and 55 the other. Smaller heads may show 21 and 34, and very large ones 89 and 144. Some heads break the pattern, so always count.

1Try It Yourself

A giant sunflower has 55 spirals turning clockwise. Its counterclockwise spirals follow the next Fibonacci number.

How many counterclockwise spirals do you expect?

Step 1: Write the mathematical expression

Add 55 and the Fibonacci number just before it:

Climbing Stairs

If you climb stairs taking either 1 or 2 steps at a time, the number of different ways to reach the top follows the Fibonacci rule.

Example:

1 stair: 1 way. 2 stairs: 2 ways (1 + 1 or 2). 3 stairs: 3 ways. To reach stair nn your last move was a 1-step from stair n−1n - 1 or a 2-step from stair n−2n - 2, so ways(nn) = ways(n−1n - 1) + ways(n−2n - 2).

2Try It Yourself

A staircase has 8 stairs. You climb 1 or 2 stairs at a time. 1 stair has 1 way, 2 stairs have 2 ways, and each next count is the sum of the two before.

In how many different ways can you climb all 8 stairs?

Step 1: Write the mathematical expression

Add the ways for 6 and 7 stairs:

Miles to Kilometers

One mile is about $1.609$ km, very close to $\varphi \approx 1.618$. So a Fibonacci number of miles is roughly the next Fibonacci number of kilometers.

Example:

5 miles ≈\approx 8 km (exactly 8.058.05 km), 8 miles ≈\approx 13 km (exactly 12.8712.87 km), 13 miles ≈\approx 21 km (exactly 20.9220.92 km).

3Try It Yourself

A cycling route is 21 miles long. You want a quick estimate in kilometers.

About how many kilometers is 21 miles?

Step 1: Write the mathematical expression

Use the next Fibonacci number after 21:

Key Takeaways

  • 1The Fibonacci sequence 1,1,2,3,5,8,13,…1, 1, 2, 3, 5, 8, 13, \ldots follows the recursive rule Fn=Fn−1+Fn−2F_n = F_{n-1} + F_{n-2} with F1=F2=1F_1 = F_2 = 1
  • 2Fibonacci's rabbit puzzle (1202) produces exactly these numbers month by month
  • 3Every third term is even (F3,F6,F9,…F_3, F_6, F_9, \ldots) and all others are odd
  • 4The sum of the first nn terms is Fn+2−1F_{n+2} - 1
  • 5The ratio Fn+1Fn\frac{F_{n+1}}{F_n} approaches the golden ratio φ=1+52≈1.618\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.618, swinging above and below it
  • 6Other starting numbers, like the Lucas numbers 2,1,3,4,7,…2, 1, 3, 4, 7, \ldots, give the same limit φ\varphi; Fibonacci numbers appear often in plants, but not in every spiral

Frequently Asked Questions

If the ratio settles at a number rr, then Fn+1Fn=1+Fn−1Fn\frac{F_{n+1}}{F_n} = 1 + \frac{F_{n-1}}{F_n} turns into r=1+1rr = 1 + \frac{1}{r}, so r2=r+1r^2 = r + 1. The positive solution is r=1+52≈1.618r = \frac{1 + \sqrt{5}}{2} \approx 1.618, the golden ratio. Check: 1.6182≈2.618=1.618+11.618^2 \approx 2.618 = 1.618 + 1.
If the ratio settles at a number rr, then Fn+1Fn=1+Fn−1Fn\frac{F_{n+1}}{F_n} = 1 + \frac{F_{n-1}}{F_n} turns into r=1+1rr = 1 + \frac{1}{r}, so r2=r+1r^2 = r + 1. The positive solution is r=1+52≈1.618r = \frac{1 + \sqrt{5}}{2} \approx 1.618, the golden ratio. Check: 1.6182≈2.618=1.618+11.618^2 \approx 2.618 = 1.618 + 1.
Yes, it is called Binet's formula: Fn=φn−(1−φ)n5F_n = \frac{\varphi^n - (1 - \varphi)^n}{\sqrt{5}}. Even though it contains 5\sqrt{5}, it always gives a whole number. Its proof comes later in algebra; for now, adding terms is the reliable way.
Both versions are common. Many books add F0=0F_0 = 0 in front: 0,1,1,2,3,5,…0, 1, 1, 2, 3, 5, \ldots With F0=0F_0 = 0 and F1=1F_1 = 1 every term keeps its number (F10=55F_{10} = 55 either way); positions only shift if someone calls that 00 the first term. In this lesson F1=F2=1F_1 = F_2 = 1.
Probably not as much as people claim. You can almost always find some pair of lengths with a ratio near 1.61.6 if you pick your lines freely, and there is little historical evidence that these artists planned it. The plant spirals are much better supported, and even there the counts are not always Fibonacci numbers.

Glossary

Fibonacci sequence
The sequence 1,1,2,3,5,8,…1, 1, 2, 3, 5, 8, \ldots where each term is the sum of the two terms before it
Recursive rule
A rule that finds each term from earlier terms, such as Fn=Fn−1+Fn−2F_n = F_{n-1} + F_{n-2}
Golden ratio (φ\varphi)
The number 1+52≈1.618\frac{1 + \sqrt{5}}{2} \approx 1.618, the limit of the ratio of neighbouring Fibonacci numbers
Lucas numbers
The sequence 2,1,3,4,7,11,…2, 1, 3, 4, 7, 11, \ldots built with the Fibonacci rule from the starting numbers 2 and 1
Golden spiral
The spiral made of quarter circles drawn in squares whose sides are Fibonacci numbers

Formula Card

Recursive rule

Fn=Fn−1+Fn−2F_n = F_{n-1} + F_{n-2}, F1=F2=1F_1 = F_2 = 1

Each term is the sum of the two before it

Sum of the first $n$ terms

F1+F2+⋯+Fn=Fn+2−1F_1 + F_2 + \cdots + F_n = F_{n+2} - 1

One less than the term two places further on

Golden ratio

φ=1+52≈1.618\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.618

The limit of $\frac{F_{n+1}}{F_n}$

Key property of $\varphi$

φ2=φ+1\varphi^2 = \varphi + 1

Squaring $\varphi$ is the same as adding 1

Today's challengeOne problem, three tries

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