# math/algebra

what is the fibonacci sequence???

1. 👍 0
2. 👎 0
3. 👁 103
1. also...wat is pascal's triangle....i need definitions...

1. 👍 0
2. 👎 0
posted by Miley

1. 👍 0
2. 👎 0
3. Leonardo Fibonacci, originally known as Leonardo of Pisa, was an Italian merchant and mathematician who contributed much to the field of algebra, Euclidian geometry, Diophantine equations, and number theory. He was instrumental in introducing the Hindu-Arabic number system to Europe. Among his many writings was the Liber Abaci, published in 1202, which contained many problems, the most famous of which, about rabbits, led to what we refer to today as Fibonacci numbers or the Fibonacci sequence. It has been quoted many ways in historical literature but basically asks, "How many pairs of rabbits can be produced from a single pair in a year, each pair producing a new pair after the second month and every month thereafter? The accumulation of rabbits looks like the following.
End of Month No.------1.....2.....3.....4.....5.....6
Pair No. 1-----------------1.....1.....1.....1.....1.....1
Pair No. 2-----------------.............1.....1.....1.....1
Pair No. 3----------------.....................1.....1.....1
Pair No. 4----------------............................1.....1
Pair No. 5---------------.............................1.....1
Pair No. 6---------------....................................1
Pair No. 7--------------.....................................1
Pair No. 8--------------.....................................1
Total........................1.....1.....2.....3.....5......8.....13.....21.....34.....55.....89.....144.....233.....377

As you can readily see, the sequence continues 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,........n, each succeeding term being the sum of the previous two terms expressed by Fn = F(n-1) + F(n-2). The initial terms are F1 = 1 and F2 = 1.

Each successive pair of Fibonacci numbers are relatively prime, i.e., they have no common factors other than 1.

Each Fibonacci number is defined in terms of the recursive relationship, Fn = [F(n-1) + F(n-2)]. To determine the 10th, 100th, or 1000th Fibonacci number, one would normally have to compute the previous 9, 99, or 999 numbers in order to compute the one desired. It is only natural therefore, to ask whether there is a simple, or complex, expression out there someplace that would allow us to calculate any Fibonacci number desired.
Search no more and surprisingly, to me at least, it involves the equally famous Golden Ratio or Golden Number,
t = (1 + sqrt5)/2, and its reciprocal, 1/t. The expression is simply

Fn = (t)^n - (-t)^-n = (t)^n - (-1/t)^n
............sqrt(5)............ sqrt(5)

where t = the famous 1.618033988749894....... or simply 1.618, as we normally use it.

The ratios of any one Fibonacci number to the previous number progressively close in on the Golden Ratio, 1.6180, = (sqrt5 + 1)/2. Surprisingly, individual terms of the Fibonacci sequence also derive from the Binet expression
Fn = [((1 + sqrt5)/2)^n - ((1 -sqrt5)/2)^n]
............................sqrt(5)

This amazingly simple expression involving square roots and powers of an irrational number does, in fact, produce the numbers in the series.

Other ways of expressing the same thing are

Fn = [(1 + sqrt5)^n - (1 - sqrt5)^n]
.......................sqrt(5)

= [t^n - (1 - t)^n]
.........sqrt5

= [t^n - (-1/t)^n]
........sqrt5

The derivation of these may be found in any good book on number theory or recreational mathematics.

Considering the first expression, lets see what we get:

n................t^n...................(-1/t)^n..............t^n - (-1/t)^n.............[t^n - (-1/t)^n]/sqrt5

1..........1.6180339............-.6180339...............2.2360679...........................1
2..........2.6180339...........+.3819660...............2.2360679...........................1
3..........4.2360679............-.2360679...............4.4721359...........................2
4..........6.8541019...........+.1458980...............6.7082039...........................3
5........11.0901699............-.0901699.............11.1803398...........................5
6........17.9442719...........+.0557280.............17.8885438...........................8
7........29.0344418............-.0344418.............29.0688837..........................13
8........46.9787137...........+.0212862.............46.9574275..........................21

How elegant.

1. 👍 0
2. 👎 0
posted by tchrwill
4. There is one famous arrangement of numbers in a familiar geometric shape that has received the attention and admiration of mathematicians for centuries, Pascal's Triangle. Contrary to popular belief, it was not created by Pascal but is believed to have been created, or discovered, by both the Chinese and Persians sometime during the 11th and 12th centuries. Blaise Pascal had the distinction of having it named after him merely because of his extensive 17th century work with it in relation to probability. Surprisingly, it has connections with probability, combinations, the binomial expansion, Taxicab Geometry, powers of 2, and perhaps many others which I have not yet had the priviledge of hearing about.

The first appearances of the triangle were alledgedly associated with the coefficients of the binomial expansion. But lets first define the array and then show how it applies. We are going to create a triangle with an array of numbers within the triangle. The apex of our triangle is the number 1 and called row 0. The following row 1 contains 2 1's. We start out as follows:
..................................................................................1
..............................................................................1......1
The next row contains a 1, 2, and 1 as in
..................................................................................1
..............................................................................1......1
...........................................................................1.....2......1
The next row contains a 1, 3, 3, and 1 as in
Row
...0...............................................................................1
...1...........................................................................1......1
...2........................................................................1.....2......1
...3.....................................................................1....3......3......1

If you do not see the evolving pattern, now is probably the best time to explain how to create the rest of the triangle. In all its simplicity, each number is simply the sum of the two numbers immediately above. Looking at the 2nd row, the 1 at the beginning of each row is the sum of the 1 and the implied 0 above it. The next number is the sum of the two 1's above it. The last one is derived the same as the first 1 in the row. The 2nd and 3rd numbers in the 3rd row are the sum of the 1's and 2's above them. Therefore, we can continue to construct the triangle as far as we wish as follows:

Row
0..................................................................................1
1..............................................................................1......1
2...........................................................................1.....2.....1
3........................................................................1....3......3.....1
4.....................................................................1....4.....6......4.....1
5..................................................................1....5...10....10.....5.....1
6...............................................................1....6...15....20....15....6.....1
7............................................................1....7...21...35...35....21.....7.....1
8.........................................................1....8...28...56...70....56...28....8.....1
9......................................................1....9...36..84..126..126..84....36.....9.....1
10.................................................1...10..45.120..210.252..210..120...45...10.....1

1. 👍 0
2. 👎 0
posted by tchrwill

## Similar Questions

1. ### Pre-Calc

Find three examples of the Fibonacci sequence in nature. Write a paragraph for each example. For each example, address the following questions: How does the example relate to the Fibonacci sequence? What portions of each item or

asked by Anonymous on September 11, 2011
2. ### Math (fibonacci

A sequence of numbers is called a Fibonacci-type sequence if each number (after for the first two) is the sum of the two numbers which precede it. For example, 1, 1, 2, 3, 5, 8 ... is a Fibonacci- type sequence. If 1985, x, y, 200

asked by Lexmo on August 8, 2014
3. ### Sequences

Like the Fibonacci sequence, a certain sequence satisfies the recurrence relation an=an−1+an−2. Unlike the Fibonacci sequence, however, the first two terms are a1=4 and a2=1. Find a32.

asked by Daniel on February 9, 2018
4. ### Maths

The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, ... starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the

asked by AMT on November 18, 2016
5. ### Math

F25= 75,025 and F26= 121,393 where Fn is the nth term in the Fibonacci sequence. Find F27. I do not understand the Fibonacci sequence, could someone help me with my question and explain it to me please.

asked by Punkie on December 6, 2009
6. ### Computer Science - MATLAB

One interesting property of a Fibonacci sequence is that the ratio of the values of adjacent members of the sequence approach a number called “the golden ratio” or PHI. Create a program that accepts the first two numbers of a

asked by George on July 8, 2011
7. ### Computer Science - MATLAB

I guess it was skipped One interesting property of a Fibonacci sequence is that the ratio of the values of adjacent members of the sequence approach a number called “the golden ratio” or PHI. Create a program that accepts the

asked by George on July 8, 2011

what is the fibonacci sequence and what is its relationship to the golden ratio? http://www.google.com/search?q=fibonacci+golden+ratio&start=0&ie=utf-8&oe=utf-8&client=firefox-a&rls=org.mozilla:en-US:official Many websites here

asked by Kelly on September 16, 2006

I am doing a math project on Leonardo Fibonacci, the creator of the Fibonacci's Sequence. I was wondering if someone could please tell me a website where I could get more information about his marriage. can someone help me?

asked by trixie on August 22, 2005
10. ### math

There is no function equation for the Fibonacci numbers, you have to use a Recursion formula. F(n) = F(n-1) + F(n-2) for n>2 and F(1)=1, F(2)=1 what is the equation for the fibonacci sequence

asked by Reiny on March 11, 2007
11. ### Maths

A super-Fibonacci sequence is a list of whole numbers with the property that, from the third term onwards, every term is the sum of all the previous terms. For example, 1, 4, 5, 10, ... How many super-Fibonacci sequences with 1

asked by Palmer on November 16, 2016

More Similar Questions