Fibonacci Sequence

The **Fibonacci sequence** is the list of numbers $$1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \ldots$$ in which every term after the first two is the **sum of the two terms before it**: $1 + 1 = 2$, $1 + 2 = 3$, $2 + 3 = 5$, $3 + 5 = 8$, and so on. ### The Recursive Rule We call the terms $F_1, F_2, F_3, \ldots$ and write: $$F_1 = 1, \quad F_2 = 1, \quad F_n = F_{n-1} + F_{n-2} \text{ for } n \geq 3$$ You will also see it written as $F(n) = F(n-1) + F(n-2)$. 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 $d$, in a geometric sequence you always multiply by the same $r$, 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 $1, 1, 2, 3, 5, 8, \ldots$ (Fibonacci's own pair had babies from the first month, so his count began $1, 2, 3, 5, \ldots$, 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**: $F_3 = 2$, $F_6 = 8$, $F_9 = 34$, $F_{12} = 144$, and all other terms are odd. - **Sum of the first $n$ terms**: $F_1 + F_2 + \cdots + F_n = F_{n+2} - 1$. For example $1 + 1 + 2 + 3 + 5 = 12 = 13 - 1$. - **Ratio of neighbours**: $\frac{F_{n+1}}{F_n}$ gets closer and closer to the **golden ratio** $$\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.618$$

Continuing the Sequence

The Fibonacci sequence starts $1, 1, 2, 3, 5, 8, 13, 21, 34, \ldots$ Find $F_{10}$, $F_{11}$ and $F_{12}$.

Find the last two known terms: $F_8 = 21$ and $F_9 = 34$ = $F_8 = 21$, $F_9 = 34$

Add them to get $F_{10}$: $F_{10} = F_9 + F_8 = 34 + 21$ = $F_{10} = 55$

Add the two newest terms to get $F_{11}$: $F_{11} = F_{10} + F_9 = 55 + 34$ = $F_{11} = 89$

Repeat once more for $F_{12}$: $F_{12} = F_{11} + F_{10} = 89 + 55$ = $F_{12} = 144$

Answer: $F_{10} = 55$, $F_{11} = 89$ and $F_{12} = 144$. Always add the two most recent terms.

Fibonacci's Rabbit Problem

One newborn pair of rabbits lives in a field in month 1. A pair is grown up after one month, and every grown-up pair has one new pair each month. No rabbit dies. How many pairs are there in month 8?

Count the first months: Month 1: one baby pair. Month 2: that pair is grown up but has no babies yet. Month 3: it has its first baby pair. = $1, 1, 2$

Find the rule: Pairs this month = pairs last month (nobody dies) + one baby pair for every pair that was alive two months ago (those are grown up now) = $R_n = R_{n-1} + R_{n-2}$

Continue to month 8: $1, 1, 2, 3, 5, 8, 13, 21$ = Month 8: $21$ pairs

Split into grown-up and baby pairs: Grown-up pairs in month 8 = all pairs of month 7 = 13; babies = grown-ups of month 7 = all pairs of month 6 = 8 = $13 + 8 = 21$

Answer: In month 8 there are $21$ pairs: $13$ grown-up pairs and $8$ baby pairs. The rabbit counts are exactly the Fibonacci numbers.

Ratios Approach the Golden Ratio

Divide each Fibonacci number by the one before it, from $\frac{5}{3}$ to $\frac{34}{21}$. What number do the ratios approach?

Compute the first two ratios: $\frac{5}{3} \approx 1.667$ and $\frac{8}{5} = 1.600$ = $1.667$, $1.600$

Compute the next two: $\frac{13}{8} = 1.625$ and $\frac{21}{13} \approx 1.615$ = $1.625$, $1.615$

One more ratio: $\frac{34}{21} \approx 1.619$ = $1.619$

Compare with $\varphi$: $\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.618$. The ratios are above, below, above, below, above $\varphi$, and the gap shrinks every time = Ratios approach $1.618$

Answer: The ratios swing above and below $\varphi \approx 1.618$ and get closer each time. By $\frac{89}{55} \approx 1.6182$ they already agree with $\varphi$ to three decimal places.

The Sum Shortcut

Find $1 + 1 + 2 + 3 + 5 + 8 + 13 + 21$ without adding all the terms one by one.

Count the terms: These are $F_1$ to $F_8$ = $n = 8$

Use the sum rule: $F_1 + \cdots + F_n = F_{n+2} - 1$, so the sum is $F_{10} - 1$ = Sum $= F_{10} - 1$

Look up $F_{10}$: $F_9 = 34$, so $F_{10} = 21 + 34 = 55$ = Sum $= 55 - 1 = 54$

Check by adding: $1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 = 54$ = Confirmed: $54$

Answer: The sum is $54$, which is $F_{10} - 1$. The sum of the first $n$ Fibonacci numbers is always one less than the term two places further on.

New Seeds: The Lucas Numbers

Use the same rule (add the two previous terms) but start with $2$ and $1$. Find the first 10 terms and the ratio of the 10th term to the 9th.

Build terms 3 to 6: $2 + 1 = 3$, $1 + 3 = 4$, $3 + 4 = 7$, $4 + 7 = 11$ = $2, 1, 3, 4, 7, 11$

Build terms 7 to 10: $7 + 11 = 18$, $11 + 18 = 29$, $18 + 29 = 47$, $29 + 47 = 76$ = $18, 29, 47, 76$

Divide the 10th term by the 9th: $\frac{76}{47} \approx 1.617$ = $\approx 1.617$

Compare with the Fibonacci ratio: Fibonacci: $\frac{55}{34} \approx 1.618$. Both are close to $\varphi \approx 1.618$ = Same limit $\varphi$

Answer: The Lucas numbers are $2, 1, 3, 4, 7, 11, 18, 29, 47, 76, \ldots$ and $\frac{76}{47} \approx 1.617$. With any two positive starting numbers, the ratio of neighbours still approaches $\varphi$.

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 $137.5°$, linked to $\varphi$) 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 $\varphi \approx 1.618$ is close to $1.609$ (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.

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

Why: 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 $2$ comes $1 + 2 = 3$, then $2 + 3 = 5$: $1, 1, 2, 3, 5, 8, \ldots$

Mistake: Miscounting positions, for example saying $F_{10} = 89$

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

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

Mistake: Thinking the ratios climb steadily up to $1.618$, or that one ratio is exactly $\varphi$

Why: The ratios jump above and below $\varphi$: $1.6$, $1.625$, $1.615$, $1.619$, ... Every ratio is a fraction, and $\varphi$ is irrational, so no ratio ever equals it.

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

Mistake: Believing every spiral and every beautiful rectangle is Fibonacci or golden

Why: A nautilus shell is a spiral, but its growth ratio is not $\varphi$. 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.

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.

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.

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.

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

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.

5 miles $\approx$ 8 km (exactly $8.05$ km), 8 miles $\approx$ 13 km (exactly $12.87$ km), 13 miles $\approx$ 21 km (exactly $20.92$ km).

The Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, \ldots$ follows the recursive rule $F_n = F_{n-1} + F_{n-2}$ with $F_1 = F_2 = 1$

Fibonacci's rabbit puzzle (1202) produces exactly these numbers month by month

Every third term is even ($F_3, F_6, F_9, \ldots$) and all others are odd

The sum of the first $n$ terms is $F_{n+2} - 1$

The ratio $\frac{F_{n+1}}{F_n}$ approaches the golden ratio $\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.618$, swinging above and below it

Other starting numbers, like the Lucas numbers $2, 1, 3, 4, 7, \ldots$, give the same limit $\varphi$; Fibonacci numbers appear often in plants, but not in every spiral

Q: Why do the ratios approach $1.618$ and not some other number?

A: If the ratio settles at a number $r$, then $\frac{F_{n+1}}{F_n} = 1 + \frac{F_{n-1}}{F_n}$ turns into $r = 1 + \frac{1}{r}$, so $r^2 = r + 1$. The positive solution is $r = \frac{1 + \sqrt{5}}{2} \approx 1.618$, the golden ratio. Check: $1.618^2 \approx 2.618 = 1.618 + 1$.

Q: Is there a formula for $F_n$ without listing all the terms?

A: Yes, it is called Binet's formula: $F_n = \frac{\varphi^n - (1 - \varphi)^n}{\sqrt{5}}$. Even though it contains $\sqrt{5}$, it always gives a whole number. Its proof comes later in algebra; for now, adding terms is the reliable way.

Q: Does the sequence start with 0 or with 1?

A: Both versions are common. Many books add $F_0 = 0$ in front: $0, 1, 1, 2, 3, 5, \ldots$ With $F_0 = 0$ and $F_1 = 1$ every term keeps its number ($F_{10} = 55$ either way); positions only shift if someone calls that $0$ the first term. In this lesson $F_1 = F_2 = 1$.

Q: Is the golden ratio really hidden in the Parthenon and the Mona Lisa?

A: Probably not as much as people claim. You can almost always find some pair of lengths with a ratio near $1.6$ 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.

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Fibonacci Sequence

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Pre-Calculus / Sequences

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 min

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