The Fibonnaci Sequence is a sequence of numbers developed byby an Italian mathematician by the name of Leonardo Fibonnacci.Given the first two numbers, f(0) and f(1), the other numbersare generated by adding the two preceding numbers. For examplef(2)=f(0)+f(1) and f(3)=f(1)+f(2) and so on.
Using Microsoft Excell, I am going to explore the ratio ofeach pair of adjacent terms in the Fibonnaci sequence and forsimilar sequences. I will begin with f(0)=1 and f(1)=1.
In the chart below, let the first column be the consecativeintegers 0-30 and the second column be f(n). Of course the FibonnaciSequence goes on to infinity, but for our purposes and for thesake of time, we are going to look at f(0) through f(30). Alson cannot be less than 0 because we do not generally talk aboutnegative numbers when discussing the nth number of a series; howeverf(n) can be thought of as a negative value and later we will experimentwith f(n) being less than 0. Let the second column be f(n) foreach intenger in the first column. Let the third column be theratio of each number with the preceding number, i.e. f(n+1)/f(n).Also let the next column be the 2nd ratio of the f(n); therefore,each value in column 4 is the ratio of each value in column 3with its preceding value. Click here to see theactual worksheet
From our chart we see that the ratio of the Fibonacci numbersgoes to 1.61803399 as n approaches infinity and the 2nd ratioapproaches 1 because each of the 1st ratios is approximately equalto the preceding value. Therefore the limit of this sequence ofFibinocci numbers is 1.6180339 as n approaches infinity.
Lets look at another sequence similar to the Fibonnaci Sequence;Again define f(n) for integer n>1 to be f(n-1)+f(n-2), butthis time let f(0)=4 and f(1)=6. Click hereto see the actual worksheet
Again the limit is 1.6180339 as n approaches infinity withthe limit of the 2nd ratio being 1 as n approaches infinity.
Now letís look at another similar sequence know as theLucas Sequence. Like before, the we define f(n) for integer n>1to be f(n-1)+f(n-2), but in the Lucas Sequence f(0)=1 and f(1)=3.Click here to see the actual worksheet
Once again the limit as n goes to infinity is 1.61803399 andthe limit of the 2nd ratio as n goes to infinity is 1.
Up until now we have only discussed cases where f(n) is positive.We can now guess that the limit as n goes to infiny is 1.61803399when f(n) is a positive number, but what if f(n) is not a positivenumber? Obviously f(0) and f(1) can't both because then f(n) wouldalways be zero and our theory wonít work. But what if f(0)or f(1) were negative. Letís experiment with f(0)= -4 andf(1)=-7. Click here to see the actual worksheet
Again we come up with the same limits as before. We see thatf(n) being negative does not change the results. We also see thatf(1) can be less than f(0) and still work.
We now can safely assume that any sequence of numbers wheref(0) and f(1) are given and both arenít equal to zero andf(n) for integer n>1 is defined as f(n-1)+f(n-2) will havea ratio whose limit is 1.61803399 and whose 2nd ratio has a limitof 1.
The value 1.61803399 is a very interesting value that is knownas the Golden Ratio. It can also be expressed as [1+sqrt(5)]/2.As discussed in EMAT3500, in addition to being the limit of thesequences discussed in this problem, the Golden Ratio is foundin many other real life situations. Credit cards are generallymade so that the ratio of the length to the width is the GoldenRatio. The ratio of a personís height to his width is theGolden Ratio. These are only a couple of examples of where thisfascinating ratio happens to occur.
The easiest way that I have found to prove that the ratios ofthese sequences will approach the Golden Ratio is as follows:
1. for n>1 by the definition of f(n) that we set for each sequence
2. by substituting for n in step 1
3. by division
4. by the rules of reciprocals
4. Let be the limit of and would be the limit of whichis the same as.
6 = sowe'll use L to denote both of them. So bysubstitution
7. So by multiplicationand addition
8. So using the quadratic formula we get
9. by arithmetic
10. because as n gets closer to infinity is positive.
Therefore, the limit of isthe Golden Ratio.
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FAQs
The Fibonacci Sequence is a series of numbers that starts with 0 and 1, and each subsequent number is the sum of the two preceding numbers. So the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.
What is the easiest way to solve the Fibonacci sequence? ›
Fibonacci Sequence = 0, 1, 1, 2, 3, 5, 8, 13, 21, …. “3” is obtained by adding the third and fourth term (1+2) and so on. For example, the next term after 21 can be found by adding 13 and 21. Therefore, the next term in the sequence is 34.
Is there a rule for the Fibonacci sequence? ›
The sequence follows the rule that each number is equal to the sum of the preceding two numbers. The Fibonacci sequence begins with the following 14 integers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 ...
What is the golden rule Fibonacci sequence? ›
The Golden Ratio is a relationship between two numbers that are next to each other in the Fibonacci sequence. When you divide the larger one by the smaller one, the answer is something close to Phi. The further you go along the Fibonacci Sequence, the closer the answers get to Phi.
Whose real name is the first few Fibonacci numbers 0 1 2 3 5 8 these numbers are named after Fibonacci? ›
They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci.
Is there a formula for Fibonacci? ›
Yes, there is a formula for finding Fibonacci numbers. Fibonacci numbers follow this formula according to which, Fn = Fn-1 + Fn-2, where Fn is the (n + 1)th term and n > 1. The first Fibonacci number is expressed as F0 = 0 and the second Fibonacci number is expressed as F1 = 1.
What is the correct Fibonacci sequence? ›
The Fibonacci sequence is the series of numbers where each number is the sum of the two preceding numbers. For example, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, …
How accurate is the Fibonacci sequence? ›
How Accurate Are Fibonacci Retracements? Some experts believe that Fibonacci retracements can forecast about 70% of market movements, especially when a specific price point is predicted.
What is the golden ratio to calculate Fibonacci sequence? ›
The golden ratio is derived by dividing each number of the Fibonacci series by its immediate predecessor. In mathematical terms, if F(n) describes the nth Fibonacci number, the quotient F(n)/ F(n-1) will approach the limit 1.618... for increasingly high values of n. This limit is better known as the golden ratio.
Why is the Fibonacci sequence so important? ›
The Fibonacci sequence is important for many reasons. In nature, the numbers and ratios in the sequence can be found in the patterns of petals of flowers, the whorls of a pine cone, and the leaves on stems. As the sequence continues, the ratios of the terms approach a number known as the golden ratio.
The patterns are not coincidental. On a practical and scientific basis, the Fibonacci patterns optimize growth processes and access to resources like sunlight, so one might then tend to think that this is all the result of natural selection.
What is Fibonacci sequence in real life? ›
The Fibonacci sequence can also be seen in the way tree branches form or split. A main trunk will grow until it produces a branch, which creates two growth points. Then, one of the new stems branches into two, while the other one lies dormant. This pattern of branching is repeated for each of the new stems.
Does the universe follow the Fibonacci sequence? ›
Believe it or not, the Fibonacci sequence can also be found in the universe! It shows up in a number of ways. Here are a few examples: The spiral arms of galaxies: Some scientists have found that the distribution of stars in the spiral arms of galaxies follows a pattern known as the Fibonacci spiral.
Why is 1.618 so important? ›
The golden ratio, approximately between 1 to 1.618, is an extremely important number to mathematicians. But when it comes to art, artists use this golden ratio because it is aesthetically pleasing. The golden ratio can be used in art and design to achieve beauty, balance, and harmony.
Who invented the Fibonacci sequence? ›
Fibonacci (born c. 1170, Pisa? —died after 1240) was a medieval Italian mathematician who wrote Liber abaci (1202; “Book of the Abacus”), the first European work on Indian and Arabian mathematics, which introduced Hindu-Arabic numerals to Europe. His name is mainly known because of the Fibonacci sequence.
What is the best Fibonacci golden ratio? ›
What is the Fibonacci sequence? The golden ratio of 1.618 – the magic number – gets translated into three percentages: 23.6%, 38.2% and 61.8%.
What is a famous Fibonacci quote? ›
If by chance I have omitted anything more or less proper or necessary, I beg forgiveness, since there is no one who is without fault and circ*mspect in all matters.
What is the Rabbit problem Fibonacci? ›
Each pair is comprised of 1 male and 1 female and no rabbits die or leave the field. This is the classic rabbit problem Fibonacci used to generate the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… Ask students to work together in pairs and estimate the number of pairs they would have at the end of one year.
Was Leonardo Fibonacci religious? ›
He grew up and was trained in North Africa, where his father worked. His major contributions to mathematics can be found in his books- Liber Abaci, Practica Geometriae, Flos, and Liber Quadrotorum. Little can be found about his Christian faith, but fruits of his faith can be seen through some of his work.
How do you solve the golden ratio of Fibonacci? ›
The golden ratio is derived by dividing each number of the Fibonacci series by its immediate predecessor. In mathematical terms, if F(n) describes the nth Fibonacci number, the quotient F(n)/ F(n-1) will approach the limit 1.618... for increasingly high values of n. This limit is better known as the golden ratio.
We put the value of n=15 to get F15=F14+F13=233+144=377 . Therefore, the 15th term in the Fibonacci sequence of numbers is 377.
for n>1 by the definition of f(n) that we set for each sequence
by substituting 
for n in step 1
by division
by the rules of reciprocals
be the limit of
and 
would be the limit of 
whichis the same as
.
=
sowe'll use L to denote both of them. So 
bysubstitution
by multiplicationand addition
by arithmetic
because as n gets closer to infinity