Mathematics - Fibonacci Sequence and the Golden Ratio

Summary

This educational video delves into the Fibonacci sequence and the Golden Ratio, explaining how each term in the sequence is the sum of the previous two. It demonstrates the convergence of ratios between successive Fibonacci numbers towards the Golden Ratio, approximately 1.618. The video also covers the mathematical derivation of the Golden Ratio through the Fibonacci sequence and introduces a formula for calculating any term in the sequence. It concludes with insights into the geometric properties of the sequence, such as the approximation of the geometric mean by middle terms for large 'n'.

Takeaways

• 🔢 The Fibonacci sequence starts with 0 and 1, with each subsequent term being the sum of the previous two.
• 🌟 The Golden Ratio is the limit of the ratios of successive terms in the Fibonacci sequence, approximately equal to 1.618.
• 📈 As 'n' increases in the Fibonacci sequence, the ratio of successive terms approaches the Golden Ratio.
• 🔄 The sequence can be used to approximate the next term by multiplying the previous term by the Golden Ratio (1.618).
• 🔄 The reciprocal of the Golden Ratio (approximately 0.618) can be used to approximate the previous term in the sequence.
• 🧮 The exact value of the Golden Ratio can be calculated using the formula ((sqrt(5) + 1) / 2), which results in 1.618.
• 🔢 A formula to calculate any term in the Fibonacci sequence is given by f_n = ((sqrt(5) + 1)^n - (-sqrt(5) + 1)^n) / (2^n * sqrt(5)).
• 🔄 The Fibonacci sequence behaves like a geometric sequence for large values of 'n', where terms are proportional to powers of the Golden Ratio.
• 🔍 The Fibonacci sequence's properties can be derived from the quadratic equation r^2 - r - 1 = 0, leading to the Golden Ratio.
• 📘 The video provides a comprehensive overview of the Fibonacci sequence and the Golden Ratio, including their mathematical derivations and applications.

Q & A

What is the Fibonacci sequence?

The Fibonacci sequence is a series of numbers where the first two terms are 0 and 1, and each subsequent term is the sum of the previous two terms.

How is the Fibonacci sequence defined mathematically?

Mathematically, the Fibonacci sequence is defined as f(n) = f(n-1) + f(n-2) for n >= 2, with initial terms f(0) = 0 and f(1) = 1.

What is the Golden Ratio?

The Golden Ratio, often denoted by the Greek letter phi (φ), is the limit of the ratios of successive terms in the Fibonacci sequence, approximately equal to 1.618033988749895.

How does the Golden Ratio relate to the Fibonacci sequence?

As the Fibonacci sequence progresses, the ratio of successive terms approaches the Golden Ratio, which is the limit of f(n) / f(n-1) as n approaches infinity.

What is the formula for calculating the nth term of the Fibonacci sequence?

The formula for calculating the nth term of the Fibonacci sequence is f(n) = ( (sqrt(5))^n - (-sqrt(5))^n ) / (2^n * sqrt(5)).

How can you estimate a term in the Fibonacci sequence using the Golden Ratio?

You can estimate a term in the Fibonacci sequence by multiplying a known term by the Golden Ratio raised to the power of the number of steps you want to advance in the sequence.

What is the significance of the number 0.618 in relation to the Fibonacci sequence?

The number 0.618 is the reciprocal of the Golden Ratio and is used to approximate the previous term in the Fibonacci sequence when n is very large.

Can you use the Golden Ratio to find terms in the Fibonacci sequence beyond the 12th term?

Yes, for large values of n, the Golden Ratio can be used to approximate terms in the Fibonacci sequence by using the formula f(n) ≈ f(n-1) * φ^(n-1).

How is the Golden Ratio derived from the Fibonacci sequence?

The Golden Ratio is derived from the Fibonacci sequence by considering the ratio of consecutive terms and taking the limit as the term number approaches infinity, which results in the quadratic equation r^2 - r - 1 = 0.

What is the geometric interpretation of the Fibonacci sequence when n is large?

When n is large, the Fibonacci sequence approximates a geometric sequence, where each term is approximately the previous term multiplied by the Golden Ratio.

Outlines

00:00

🔢 Introduction to Fibonacci Sequence

This paragraph introduces the Fibonacci sequence, a series of numbers where each term is the sum of the two preceding ones, starting from 0 and 1. The sequence is defined mathematically as f(n) = f(n-1) + f(n-2) for n ≥ 2. The video script demonstrates the calculation of the first few terms of the sequence, illustrating how each term is derived from the sum of the previous two. The sequence's pattern is shown up to the 12th term, which is 144, and the 13th term, which is 233.

05:00

🌟 The Golden Ratio and Its Relation to Fibonacci

The golden ratio, approximately 1.618, is discussed as the limit of the ratios of successive Fibonacci numbers. The video script explains how the ratio of consecutive Fibonacci terms converges towards the golden ratio as the terms increase in value. The script also shows how the golden ratio can be used to approximate the next term in the sequence by multiplying the previous term by the ratio. The exact values of the golden ratio and its conjugate, 0.618, are derived from the square root of 5, either by adding or subtracting 1 and then dividing by 2.

10:05

🔄 Estimating Fibonacci Terms Using the Golden Ratio

The paragraph explains how to estimate higher Fibonacci numbers using the golden ratio. It demonstrates that as 'n' becomes large, the Fibonacci sequence starts to resemble a geometric sequence, where each term is approximately the previous term multiplied by the golden ratio. The video script provides a method to estimate the 20th term of the sequence by raising the golden ratio to the power of the difference in indices and multiplying it by the 12th term, resulting in an approximation of the 20th term.

15:06

📐 Fibonacci Formula and Geometric Mean

This section introduces a formula for calculating the exact value of any term in the Fibonacci sequence, which involves the golden ratio and the square root of 5. The formula is f(n) = ( (1 + √5) / 2 )^n - ( (1 - √5) / 2 )^n. The video script also discusses the geometric mean property of the Fibonacci sequence, where a term is approximately the geometric mean of its neighbors when 'n' is large. The script provides examples of calculating the 20th and 16th terms using this formula and confirms their accuracy by adding the preceding terms.

20:06

🔄 Deriving the Golden Ratio from Fibonacci

The final paragraph delves into the mathematical derivation of the golden ratio from the properties of the Fibonacci sequence. It starts with the characteristic equation of the sequence, f(n) = f(n-1) + f(n-2), and manipulates it algebraically to form a quadratic equation. The quadratic formula is then applied to solve for the golden ratio, resulting in two solutions: the golden ratio (1.618) and its conjugate (0.618). The paragraph also explains how to derive the reciprocal of the golden ratio, which is 0.618, by rationalizing the expression for the golden ratio.

Keywords

💡Fibonacci Sequence

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. It is central to the video's theme as it sets the stage for discussing the Golden Ratio. In the script, the Fibonacci sequence is used to demonstrate how each term builds upon the sum of the two previous terms, as illustrated by the sequence starting with 0, 1, 1, 2, 3, 5, and so on.

💡Golden Ratio

The Golden Ratio, often symbolized by the Greek letter phi (φ), is an irrational number approximately equal to 1.618033988749895. It is the ratio of two quantities where the ratio of the sum to the larger quantity is the same as the ratio of the larger quantity to the smaller one. In the video, the Golden Ratio is shown to be the limit of the ratios of successive terms in the Fibonacci sequence, highlighting its significance in mathematics and aesthetics.

💡Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The video explains that as n becomes large, the Fibonacci sequence approximates a geometric sequence, with the Golden Ratio acting as the common ratio, which helps in estimating the next term in the sequence.

💡Quadratic Equation

A quadratic equation is a polynomial equation of the second degree, typically written in the form ax^2 + bx + c = 0. In the video, the derivation of the Golden Ratio involves setting up a quadratic equation based on the properties of the Fibonacci sequence, which is then solved to find the values of the Golden Ratio.

💡Conjugate

In mathematics, a conjugate of a binomial is formed by changing the sign of one term. In the video, the concept of conjugates is used in the process of rationalizing the denominator to find the exact value of the Golden Ratio, which is an important step in the mathematical derivation.

💡Reciprocal

The reciprocal of a number is what you multiply that number by to get 1. In the context of the video, the reciprocal of the Golden Ratio (approximately 0.618) is derived and discussed, showing the relationship between the two values and how one can be obtained from the other.

💡Geometric Mean

The geometric mean of two numbers is the square root of their product. In the video, it is mentioned that a number in the Fibonacci sequence, when n is large, approximates the geometric mean of the previous and successive terms, illustrating the close relationship between the Fibonacci sequence and geometric progressions.

💡Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. The video uses the square root of 5 in the context of deriving the Golden Ratio, showing that the ratio can be expressed in terms of the square root of 5 plus or minus 1, divided by 2.

💡Approximation

Approximation in mathematics refers to representing a number or value with a close but not exact value. The video discusses how the Golden Ratio can be used to approximate the next term in the Fibonacci sequence, especially when dealing with large numbers, showcasing the practical use of mathematical concepts.

💡Fibonacci Formula

The Fibonacci formula, also known as Binet's formula, is used to calculate the nth Fibonacci number directly without computing the previous numbers in the sequence. The video provides the formula as an alternative method to find Fibonacci numbers, demonstrating its utility in simplifying calculations.

Highlights

The Fibonacci sequence starts with 0 and 1, with each subsequent term being the sum of the previous two.

The nth Fibonacci number (F(n)) is defined as the sum of the two preceding numbers for n >= 2.

The golden ratio is the limit of the ratios of successive Fibonacci terms as n approaches infinity.

As n increases, the ratio of successive Fibonacci numbers converges to approximately 1.618.

The golden ratio can be used to estimate the next term in the Fibonacci sequence when n is large.

The exact value of the golden ratio is derived from the square root of 5, plus or minus one, divided by 2.

The reciprocal of the golden ratio (approximately 0.618) can be used to approximate previous terms in the sequence.

For large n, the nth Fibonacci number is approximately the (n-1)th number multiplied by the golden ratio.

The Fibonacci sequence can be used to calculate the exact value of any term using a specific formula involving the golden ratio.

The Fibonacci sequence exhibits a property where each term is approximately the geometric mean of its neighbors for large n.

The golden ratio can be derived from the Fibonacci sequence by setting up a quadratic equation.

The quadratic formula is used to find the golden ratio from the derived quadratic equation.

The golden ratio and its reciprocal are the two solutions to the quadratic equation r^2 - r - 1 = 0.

The Fibonacci sequence's relationship with the golden ratio can be used to approximate terms without direct calculation.

The video provides a method to calculate the exact value of any term in the Fibonacci sequence using a formula.

The video concludes with a discussion on the derivation of the golden ratio and its reciprocal from the Fibonacci sequence.

Transcripts

in this video we're going to focus on

the fibonacci sequence and the golden

ratio

so let's talk about the fibonacci

sequence first

the first term in the fibonacci sequence

is 0

the next term is 1

and

each successive term is going to be the

sum

of the previous two terms

so fn is going to be the sum of

fn minus 2 and f sub n minus 1 where n

is equal to or greater than 2.

so the first two numbers in the

fibonacci sequence are 0

and 1. to get the next number you need

to add these two numbers up 0 plus 1 is

one

and then to get the next number add the

previous two numbers

one plus one is two

one plus two is three

two plus three

is five

three plus five is eight

five plus eight is thirteen

eight plus thirteen is twenty one

thirteen plus twenty one is thirty four

twenty one plus thirty four is fifty

five

thirty 34 plus 55 is 89

and so forth

so you'll get these numbers as well

i'm going to stop at 377.

so f sub 0

is a zero

that's that the first term

f sub 1 is the next term

f sub 5

is 5.

this is f sub 10.

which really is the 11th term if you

think about it because f sub 5 is the

sixth term

i just want to make sure that you know

what these numbers represent

but now let's talk more

about the fibonacci sequence

and the golden ratio

you might be wondering what is the

golden ratio

the golden ratio

is the limit of the ratios of successive

terms in the fibonacci sequence

so if we take

the third term and divided by the second

term

that is one divided by one we're going

to get one

if we take

f sub 3 divided by f sub 2 or take the

fourth term divided by the third term

that's going to be 2 over 1 which is 2.

and then if we take 3

and divided by the previous term 2 we

get 1.5

if we take 5 and divided by 3

we're going to get

so let me get my calculator out for this

one

so this is 1.6 repeating or 1.667 if you

round it

next let's take 13 and divide it by 8.

notice

what these numbers approach

so this is going to be 1.625

and then if we take 21 divided by 13

that's going to be

1.61538 and then let's take 34

divided by 21

so that's going to be

1.619047

and then

55

divided by 34.

this is going to be 1.6176

approximately

and then 89

divided by 55.

so

looking at

well this is going to be 1.618

repeating

so looking at these numbers

as this sequence progresses

what is the ratio of the

and what is the ratio of the successive

term by the previous term

notice that it's approaching 1.618

if we try a few more let's say if we

take

144 and divided by 89

you could see this pattern develop even

further this is going to be 1.617978

and then the next one

233 divided by 144

this is

1.61805 repeating

so the golden ratio is approximately

1.618

that's not the exact answer but it

rounds to that value

so let's think about what this means

as n becomes sufficiently large

the fibonacci sequence

approaches or approximates a geometric

sequence so

starting with the number 144

if we multiply 144

by 1.618

we can get 233

if you type in 144 times 1.618

it'll give you

232.992

which is approximately 233

and then if you take 233

multiply by 1.618

you can get the next number

this would be

376.994

which rounds

to

377.

so if we take 377 multiplied by 1.618

we can approximate the next number in

the fibonacci sequence

and that is going to be

609.986 which is approximately 610

and you can confirm that if you add the

previous two numbers

233 plus 377 it gives you 610

so the golden ratio helps us

to get the next term in the sequence

now you can also go backwards

if you take 233 and

multiply it by 0.618

it'll give you 144.

you'll get

143.994 likewise if you take 144

multiply by

0.618

it'll give you approximately the

previous term when n is very large

this will give you 88.992 which rounds

to 89.

now if you try to use that for the

smaller numbers here it's not going to

work very well so when n is let's say

greater than 12

then you could use the golden ratio to

approximate the next term in the

sequence

now what do we get these numbers 0.618

and

1.618 what do they come from

here's the exact values of those two

numbers

if you type in the square root of five

and then add one to it

and then divide by two

you're going to get the first number the

larger one

which is point six one eight

zero

three three nine eight nine

and then if you take the square root of

five subtract it by one

and divide by two

you're going to get point

six one eight

zero three

three nine eight eight seven

now late in this video i'm going to talk

about how you can actually

get those exact values but that's where

the numbers come from so that's the

exact value of the golden ratio

so here's a question for you

we know that f sub 12 is one forty four

as we said before this is f sub zero f

sub one

f sub two

this is f sub six

f sub ten

so f sub 12 is 144.

knowing that what is the value of f sub

20

how can you estimate

f sub 20 or even calculate

the exact value

instead of just adding numbers in a

sequence

well keep in mind when n is very large

the fibonacci sequence approaches a

geometric sequence

so we could say that f sub 20

is approximately

f sub 12

times the golden ratio

since we're increasing we're going to

use the square root of 5 plus 1

divided by 2

raised to the 8th power

because

12 plus 8 is 20.

we need to multiply f sub 12 by the

golden ratio 8 times to get to f sub 20.

so go ahead and type that in so this is

going to be

144

times

for those of you who want a decimal

value this is one point six one eight

zero three three

nine eight nine

raised to the eighth power

so you should get

6764.935 approximately

surrounding that to the nearest whole

number

this is 64.65

so that's the 20th term in the sequence

now let's check it to make sure that

this is correct

if we add these two terms

2 33 plus 377 we're going to get that

number that we had before which is 610

and then if we have 377 plus 610

that's 987

and then 610 plus 987

that's 15

97

and then adding that to 987 that's

25.84

and then 1597 plus 2584

that's 41 81

and then 41 81 times or plus 2584

that gives us 67

65.

this is f sub 14

f17 and this is f20

and so we can see that this answer is

indeed correct

now it turns out that

there's a formula

where you can calculate the exact value

of this number

and here it is

f sub n

is

equal to

one

plus the square root of five

raised to the n power

and then it's minus

one minus the square root of five

raised to the end

divided by

2 to the n times the square root of 5.

so if we want to calculate f sub 20

it's going to be

1 plus the square root of 5. raised to

the 20th power

minus 1 minus the square root of 5

raised to the 20th power

divided by 2

raised to the 20th power times the

square root of 5.

now

you could use a scientific calculator to

get

the exact answer

but for those of you who don't have it

you may have to use some decimal numbers

so once you type it in correctly you

should get

67.65

and it gives you the exact value

so we could try another one

let's say if we want to calculate

f sub 16 it's going to be 1

plus the square root of 5 raised to the

16

minus 1 minus the square root of 5

raised to the 16

divided by 2 raised to the 16

times the square root of 5. you may want

to put that in parentheses

and this is equal to 987

which is the number that we see here

so that's the formula

that you could use to calculate any

number in the fibonacci sequence

now there are some other interesting

things regarding this sequence

that we could talk about

for instance let's focus on this number

f sub 13.

f sub 13 is between f sub 12 and f sub

14.

in fact f sub 13

is the geometric mean

or it's approximately rather

it approximates the uh

it's approximately the geometric mean

between f12 and f14

since the fibonacci sequence

approximates a geometric sequence when

ed when n is large

so if you were to take

the square root of 144

multiplied it by the square root of 377

this will give you

232.997

which is approximately

233 the middle number

so when n is large

the middle number approximates the

geometric mean of

the previous term and the successive

term

now you could try that with another

number

for instance

if you take the square root

of let's say

987 multiplied by the square root of

2584

you'll get 15 96.9997

which is approximately 1597.

now let's talk about how we can

derive the values of the golden ratio

this number 1.618 and the other number

0.618

we're going to start with this

f raised to the n

is equal to f raised to the n minus 1

plus f raised to the n

minus 2.

dividing both sides by

f to the n

we're going to get 1

which is equal to

f raised to the n minus 1 divided by f

to the n

whenever you divide you need to subtract

the exponents

so n minus 1

minus n

that's n minus 1 minus and these cancel

you just get negative 1.

so this is f to the negative one

the same is true here n minus two minus

n will give us negative two so we get f

raised to the negative two

now i'm going to multiply every term by

f squared

so this is going to be f squared is

equal to

f to the negative 1 times f squared you

need to add the exponent to negative 1

plus 2 is 1.

so you get f to the first power

and negative two plus two is zero

so that gives you f raised to the zero

power anything raised to zero power is

one

so we have this

moving f and one to the other side

we get

f squared minus f minus one is equal to

zero

now let's talk about how we could derive

the numbers that describe the golden

ratio 1.618

and 0.618

so we know that

each number in the fibonacci sequence is

the sum

of

the previous two numbers

f sub n

is going to be proportional to r to the

n

the reason for that

when we had numbers like 55

89

144

233

377

if we multiply 144

by the golden ratio which we'll call r

1.618

it's going to give us the next number

233 and if you multiply 233 by r or

1.618 it will give you approximately the

next number

so we could say that f sub

f sub n

is proportional to r to the n

and the previous term is going to be

proportional to r raised to the n minus

1.

and the previous term to that will be

proportional to r raised to the n minus

2.

so we're going to start with this

equation

r to the n is equal to r to the n minus

1 plus r

raised to the n minus 2.

now what we're going to do is we're

going to divide each term by

r raised to the n

so on the left these two will cancel

giving us one

now r to the n minus one divided by r to

the n we need to subtract the exponents

n minus one

minus n the n's will cancel giving us r

to the negative one

and dividing these two we're gonna get

r to the minus two

now the next thing we're gonna do is

multiply everything by r squared

so r squared times one is just

r squared

r squared times r to the negative one we

need to add the exponents negative one

plus two is one

and then

negative two plus two is zero

anything raised to zero power is just

one

now we're going to take the two terms on

the right side and move it to the left

side

so we're gonna have r squared

minus r minus one is equal to zero

and so what we have is an equation in

quadratic form

so we're going to use the quadratic

formula to get the answer

r is going to be negative b plus or

minus the square root

of b squared minus 4ac

divided by 2a

and this is in the form a r squared

plus br plus c

so we can see that a

a is one

b

that's a negative one

and c

is also negative one

so it's negative and then b is negative

one

plus or minus the square root of

negative one squared minus four

a is positive one

c is negative one divided by two times

one

so these two negatives will cancel

that's gonna become positive one

negative 1 squared is positive 1

and then we have negative 4 times

negative 1 which is positive 4

divided by

2. so we're going to get 1 plus or minus

1 plus 4 is 5 divided by 2.

and so thus we have

the two values for

the golden ratio

so we have

1 plus the square root of 5 or we could

say the square root of 5 plus 1 divided

by 2 which gives us

1.618

and then 1 minus the square root of 5.

that's going to give us a negative

number

which is negative 0.618

which we really don't want that

what we do want is the square root of 5

minus 1 over 2

and that will give us 0.618

at least you know how to get this number

and this one to just the negative

version of it

now there is another way you can get

this number from this number

and

it's by taking the reciprocal of it

if you take one and divide it by

1.618

you're going to get

.618

so

if you were to take this number and put

it

under one

here's how you can get the other number

or at least the exact value of the other

number

so that's what we have right now

so first let's multiply

the top and the bottom by two

so that these twos will cancel

and we're going to get 2 divided by the

square root of 5 plus 1.

our next step is to multiply the top and

bottom by the conjugate

of the denominator which is the square

root of 5 minus 1.

so on the numerator we're going to get 2

times the square root of 5 minus 1

and on the denominator we need to foil

the square root of 5 times the square

root of 5

is the square root of 25

which is 5.

and then this is going to give us

negative root 5

and then positive root 5

and then 1 times negative 1 is negative

1.

the two middle terms will cancel

they will add up to 0 and then we'll

have

minus one which is four

so we have two times the square root of

five minus one over four

now working towards the left

we can break down four into two times

two

and cancel a two

and so this gives us the other number

the square root of five minus one over

two

which is

0.618

so those are some different ways in

which you can get

these two golden ratios 1.618

and 0.618

or these two numbers

so that's it for this video hopefully it

gave you uh

some good information on the fibonacci

sequence and the golden ratio thanks for

watching