The origin of the Fibonacci Sequence

The Fibonacci sequence is a series of numbers where a number is found by adding the two previous numbers in the series.  Starting with 0 and 1, the sequence is …

Term X0 X1 X2 X3 X4 X5 X6 X7 X8 X9
Value 0 1 1 2 3 5 8 13 21 34

The 2 is found by adding the two numbers before it ( 1+ 1)

The 3 is also found by adding the two numbers before it (2 + 1)

The 5 is found by adding the two numbers before it (3 + 2)  etc …

If we were to continue this sequence shown above the next number would be 34 + 21 = 55.

The Rule

We can write a “Rule” for the Fibonacci sequence that would explain how we get the numbers in the series.

The Rule is Xn = Xn-1 + Xn-2

where:

Xn is term number “n”

Xn-1 is the previous term (n – 1)

Xn-2 is the term before that (n – 2)

For example the 7th term is calculate with the 6th and 5th terms

x7= x6 + x5

13 = 8 + 5

In Nature

Fibonacci numbers (the numbers found in the Fibonacci sequence) can be found in nature.  If you look at flowers carefully you will notice that most flowers have a Fibonacci number of petals.  The lily and the iris have three petals, the pink and the buttercup five, the delphinium has eight petals, the marigold 13, the aster 21 and daisies either 55 or 89.

250px-Creeping_butercup_close_800 Buttercup flowers have 5 petals

Pinecones, pineapples,  cauliflower, and sunflowers also have Fibonacci numbers in spiral arrangements.  In these cases you can count spirals clockwise and counter clockwise. The number of spirals you can count in both directions are consecutive Fibonacci numbers.  Pineapples typically have 5 and 8 spirals, or 8 and 13 spirals.  Spruce cones tend to have 8 and 13 spirals.  Sunflowers can have 21 and 34 , or 34 and 55 spirals- sometimes they can have as many has 144 and 233.

Lesson_Fibonacci-pineconesFibonacci numbers as seen in pinecones

main-qimg-f9fd1850f0fa3cbb12f2cef0edec68b3In pineapples

It’s Origin

The Fibonacci sequence got its name because the terms appear in mathematician Fibonacci’s book Liber Abaci, in a problem about rabbits.  It should be noted that the sequence was named “The Fibonacci Sequence” more than 600 years after the book was published when in 1877, when Edouard Lucas (a number theorist) was studying the sequence found in the book and decided to pay tribute to Fibonacci by naming the sequence after him.

The sequence was set up in Liber Abaci in the following way: Say we have a pair of rabbits, and after one month, the pair gives birth to another pair.  If every adult pair of rabbits give birth to a pair of baby rabbits every month, and it takes a month for the baby rabbits to become adults, how many rabbits are produced from the first pair in a year?

We can find the answer to this question by counting rabbits month by month.

  • In the first month there is just one pair
  • In the second month there are two pairs, as the original pair has given birth to a pair.
  • In the third month there are three pairs, the original has again bred another pair, and pair they gave birth to in the second month become adults.
  • In the forth month the two adult pairs breed, adding  two more pairs to the population. Giving us a total of five pairs.

The Fibonacci sequence is the month to month total of pairs:

  Total Pairs
First month: 1 adult pair 1
Second month: 1 adult pair and 1 baby pair 2
Third month: 2 adult pairs and 1 baby pair 3
Forth month: 3 adult pairs and 2 baby pairs 5
Fifth month: 5 adult pairs and 3 baby pairs 8
Sixth month: 8 adult pairs and 5 baby pairs 13
Seventh month: 13 adult pairs and 8 baby pairs 21
Eighth month: 21 adult pairs and 13 baby pairs 34
Ninth month: 34 adult pairs and 21 baby pairs 55
Tenth month: 55 adult pairs and 34 baby pairs 89
Eleventh month: 89 adult pairs and 55 baby pairs 144
Twelfth month: 144 adult pairs 89 baby pairs 233

So there you have it the answer to the question is 232 pairs of rabbits are produced from the first pair in a year.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s