what is the fibonacci sequence???
also...wat is pascal's triangle....i need definitions...
Google these terms.
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.
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
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. The sequence starts with 0 and 1, and each subsequent number is the sum of the previous two. Therefore, the sequence goes like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.
To compute the Fibonacci sequence, you can either iterate through each number and calculate it based on the previous two numbers, or use a mathematical formula that directly gives you the value of the nth Fibonacci number.
To compute the Fibonacci sequence using iteration, you can start with the first two numbers, 0 and 1, and repeatedly add the last two numbers to obtain the next number in the sequence.
To compute the nth Fibonacci number using a mathematical formula, you can use Binet's formula, which is:
Fn = (phi^n - (1-phi)^n) / sqrt(5)
where "Fn" is the nth Fibonacci number, "phi" is the golden ratio (√5 + 1)/2, and "^" denotes exponentiation.
Using either of these methods, you can generate the Fibonacci sequence or find specific Fibonacci numbers.