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 | X_{0} | X_{1} | X_{2} | X_{3} | X_{4} | X_{5} | X_{6} | X_{7} | X_{8} | X_{9} |
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
x_{7}= x_{6 }+ x_{5}
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.
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.
Fibonacci numbers as seen in pinecones
In 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.