The Fibonacci Sequence

The first two terms of the Fibonacci sequence are 1,1. From then on, each term is the sum of the previous two terms. This makes the third term 1 + 1 = 2, and the fourth term 1 + 2 = 3, and so on. The first ten Fibonacci numbers are shown below:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55

A classic "real life" example of Fibonacci numbers is with rabbits. It goes a bit like this: I start with a pair of young rabbits in month 1. For their first month, they do not breed, as they are too young. However, for every month after this, they produce one new pair of young rabbits, which in turn grow up in a month and start breeding. If we call a junior pair of rabbits JJ and a senior pair SS, this is what happens:

Month

1. JJ = 1 pair

2. SS = 1 pair

3. SS + JJ = 2 pairs

4. SS + SS + JJ = 3 pairs

5. SS + SS + SS + JJ + JJ = 5 pairs

6. SS + SS + SS + SS + SS + JJ + JJ + JJ = 8 pairs

The number of pairs of rabbits gives the Fibonacci numbers! Luckily, in reality, there are certain factors that limit this growth in population. The rabbits will eventually die and stop producing new pairs, there will be limited food and space, etc. According to the sequence, by the end of two years, I would have over 46,000 pairs of rabbits, so it is a good thing that this is not really what happens in nature!

However, the Fibonacci sequence does have some closer links with nature. For example, flowers tend to have a Fibonacci number of petals. This is why it's so hard to find a four leaved clover! If you ever get really bored, you can count the number of petals on a daisy - it may not be exact, but it should be around 55 or 89! Other links with nature include the arrangement of seeds in sunflowers and pines in pine cones, and even the spirals of the shells of some animals.

One final interesting property of Fibonacci numbers is that they can be found in Pascal's triangle. They are perhaps the hardest pattern to spot of all the commonly discussed sequences that can be found in Pascal's triangle, but they are worth searching long and hard for, because they are one of the amazing patterns that can be found in Pascal's triangle! (If you want a hint, then don't look for the numbers themselves, but try adding together numbers from the triangle to make them).