Help Me Out Please
- Thread starter TheMan
- Start date
1, 1, 2, 3, 5, 8, 13,
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.
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.
