Fibonacci Numbers

by Nikhat Parveen, UGA

Fibonacci is
perhaps best known for discovering a series of numbers that we now refer to as
the Fibonacci numbers.

1, 1, 2,
3, 5, 8, 13, 21, 34, 55, 89, 144, 233,...

This sequence was
introduced in Fibonacci's first book*, Liber abbaci*, where it arose
in the solution to a mathematical problem:

Solution:

At month zero, there is one pair of rabbits. At the beginning of the first month (January 1st), there is one pair of rabbits that has mated, but not yet given birth. At the beginning of the second month, the original pair gives birth, giving rise to another pair of rabbits...and so on. After one full year, there are 377 pairs of rabbits.

The
Fibonacci numbers form what we call a
recurrent sequence.
Beginning with 1, each term of the Fibonacci sequence
is the sum of the *two preceding numbers*.

0+1=1
1+1=2
1+2=3
2+3=5
3+5=8
5+8=13

Here is a recurrence relation
for the Fibonacci numbers:

This
equation identifies a situation, when *n* is greater than 2, where each
term of a sequence is the sum of the two preceding terms, as is the case with
the Fibonacci series.

To find a
given Fibonacci number, one must find all of the *preceding* Fibonacci
numbers. As *n* gets larger, finding the Fibonacci number becomes more
difficult. It is possible to express the *n*th Fibonacci number using an
equation. The formula for finding the *n*th term of the Fibonacci series
was discovered by Jacques Philippe
Marie Binet,
in 1843

__
Binet
Formula__

__
__

Figure
credit:

http://library.thinkquest.org/27890/theSeries3.html

Where F(x) is the Fibonacci number and x denotes the numerical order of the Fibonacci number.

If the
number that directly precedes a Fibonacci number, x, is known, it is possible to
determine x without using the Binet formula. By using the Successor formula, we
can calculate the "successor" to the term that we know.

__Successor Formula__:

Figure
credit:

http://library.thinkquest.org/27890/theSeries4.html

It is
possible to find the sum of the first 5 Fibonacci numbers for example, by adding
each term individually.

1 + 1 + 2 + 3 + 5 =
12

Alternatively, this relationship can be expressed
as an equation. This makes it easier to find the aggregate of the first *n*
Fibonacci numbers, especially for large values of *n*.

sum of first n Fibonacci numbers =
S(*n*) = F(*n*+2) - F(2)

for *n*=5:

S(5) = F(5+2) -
F(2)

= F(7) - F(2)

= 13 - 1

= 12

References:

Garland,
Trudi Hammel. 1987. Fascinating Fibonaccis: Mystery and Magic in Numbers. Dale
Seymour Publications, Palo Alto.
103 pp.

Newton,
Lynn D. 1987. Fibonacci and Nature: Mathematics investigations for schools.
Mathematics in School. 16:5:2-8.

http://library.thinkquest.org/27890/splash.html

http://pass.maths.org/issue3/fibonacci/#rabbit-answer

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