The Fibonacci sequence is quite simply the most famous and elegant (and also our favorite!) number sequence in Mathematics. But what exactly is it?
The Fibonacci sequence is a sequence (not a series; a series is a summation of the sequence) of numbers which follows a specific linear recurrence equation. Don’t worry if that sounds complicated. A linear recurrence equation is an equation that expresses a particular number of a series as a linear combination of the other numbers of the sequence (or in more formal language, as a linear combination of first degree polynomials of the sequence).
For the Fibonacci sequence, if F(n) is the nth term of the sequence, then the linear recurrence equation is F(n) = F(n-1) + F(n-2) for n >= 2. In modern terminology, F(0) is considered as 0, however in old times, F (1) was the starting point of the sequence and F (1) was equal to 1. We will use the modern convention for our article where F(0) = 0 and F(1) = 1. In simpler words, the nth term in the sequence is equal to the sum of the two terms preceding it. Using the recurrence formula, we find that the terms are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 … . Note that the sequence doesn't end at 377, it goes on and on. Now one might ask just what’ so special about this sequence as compared to just any other ordinary sequence? We will delve into that too but first something about the origin of the sequence.
Even though the sequence is named after the Italian mathematician Fibonacci (Leonardo Binacci), the sequence was discovered in India by various people in different time periods. There are several stories about the sequence’s origin. However the fact that it originated in India is undisputed. Fibonacci introduced the sequence to the western world (Europe) via his extremely famous book Liber Abaci and the concept was extremely well received by the Europeans.
Fibonacci presented the series in the form of a hypothetical problem – ‘Assuming that a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?
After solving the problem, it turns out that the number of pairs of rabbits each month is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.. i.e the Fibonacci sequence!
Now that we have covered the history of this sequence, let’s get into the main stuff. What makes the Fibonacci sequence the ‘celebrity’ of the Mathematical world?
First of all, a little exercise for you. Perform the operation F(n+1) / F(n) for n >= 0. This means divide the (n + 1)th Fibonacci term with the (n)th Fibonacci term. You will observe that as n keeps on getting larger, the ratio starts tending to 1.618. This isn’t just any random number; this is the Golden Ratio, also represented by the Greek letter φ (phi). Mathematically, two numbers are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The Golden Ratio has immense historical significance. Many famous pieces of architecture have been built using the principle of the Golden Ratio. It is said that objects following the Golden Ratio are more elegant and aesthetically pleasing. This seemingly insignificant number is pretty much everywhere! Just look it up and you will find a wealth of information about it.
Besides it’s relation with the Golden Ratio, the sequence hides in it a wealth of information that make it even more elegant and magical.
Here are some interesting observations (some of them we discovered on our own):
1. (F(n))^2 + (F(n+1))^2 = F(2n + 1)
2. F(0) + F(1) + F(2) + F(3) + …. F(n) = F(n+2) - 1
3. (F(0))^2 + (F(1))^2 + (F(2))^2 + (F(3))^2 + …. (F(n))^2 = F(n) F(n+1)
4. (F(n))^3 + (F(n+1))^3 = F(3n) + (n-2)^3 (For n > 2)
5. The digital root of the sequence starts repeating after 24 entries and the pattern that it follows is 1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9. We didn't calculate the digital roots by hand. That will be too tedious! Using a simple Java program can do the trick for you.
These are just some of the patterns. You can find many more by just sitting down and pondering over the sequence. That’s the beauty of the Fibonacci sequence. You never know what you will find!