Mathematics - Fibonacci Sequence and the Golden Ratio
Summary
TLDRThis 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
- đą The Fibonacci sequence starts with 0 and 1, with each subsequent term being the sum of the previous two.
- đ The Golden Ratio is the limit of the ratios of successive terms in the Fibonacci sequence, approximately equal to 1.618.
- đ As 'n' increases in the Fibonacci sequence, the ratio of successive terms approaches the Golden Ratio.
- đ The sequence can be used to approximate the next term by multiplying the previous term by the Golden Ratio (1.618).
- đ The reciprocal of the Golden Ratio (approximately 0.618) can be used to approximate the previous term in the sequence.
- 𧟠The exact value of the Golden Ratio can be calculated using the formula ((sqrt(5) + 1) / 2), which results in 1.618.
- đą 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)).
- đ The Fibonacci sequence behaves like a geometric sequence for large values of 'n', where terms are proportional to powers of the Golden Ratio.
- đ The Fibonacci sequence's properties can be derived from the quadratic equation r^2 - r - 1 = 0, leading to the Golden Ratio.
- đ 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 (Ï), 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) â f(n-1) * Ï^(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
đą 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 â„ 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.
đ 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.
đ 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.
đ 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 + â5) / 2 )^n - ( (1 - â5) / 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.
đ 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.
Mindmap
Keywords
đĄFibonacci Sequence
đĄGolden Ratio
đĄGeometric Sequence
đĄQuadratic Equation
đĄConjugate
đĄReciprocal
đĄGeometric Mean
đĄSquare Root
đĄApproximation
đĄFibonacci Formula
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
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