Introducing the Fibonacci Sequence


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"Introducing Fibonacci Numbers" by Mike DeHaan

“Introducing Fibonacci Numbers” by Mike DeHaan

The phrase, “the Fibonacci sequence” usually refers to a set of numbers that starts as {0, 1, 1, 2, 3, 5, 8, 13, 21…}. At least, “The Life and Numbers of Fibonacci” starts with “zero and one”. Many others would skip the zero and simply start with “one and one”, but that does not matter.

This sequence is created by following two rules:

  • The first two numbers are 0 and 1.
  • The next number is the sum of the two most recent numbers.

Actually, one must perform rule #2 over and over again, until one runs out of time, patience, paper or ink in the pen.

So, constructing the Fibonacci sequence starts from 1=1+0, then 2=1+1, then 3=2+1, then 5=3+2, 8=5+3, and so on.

Why is the Fibonacci Sequence Important?

As with many mathematical concepts, the Fibonacci sequence started as the solution to a contest. Then people began to realize it crops up in many places.

Leonardo Fibonacci Won a Contest

"Leonardo Fibonacci" Image by zoonabar

“Leonardo Fibonacci” by zoonabar

Leonardo Fibonacci invented his sequence to win a competition sponsored by Emperor Frederick II in 1225. The contest question was: Start with a pair of rabbits. Every month, every pair of rabbits who are over a month old gives birth to a new pair of rabbits. After “n” months, how many pairs of rabbits are there?

(It seems there must have been some more rules. These rabbits never die; they never get too old to reproduce; they form stable monogamous “married pairs” for life; and all births are synchronized on the first day of each month).

This problem can be solved by “mathematical induction”. The first step of this induction begins at the beginning. Throughout the first month there is only the one initial pair. At the start of the second month, one new pair is born, so there are two pairs. At the start of the third month, only the first pair has another pair, so the total is three pairs. But to kick off the fourth month, the oldest children produces a pair of rabbits; so do their parents. Adding these two new pairs to the existing three pairs, we have five pairs. So the pattern has begun: (1), 1, 2, 3, 5…

For the second part of this mathematical induction, we can say there are x pairs after (n-1] months, x[n] pairs after (n) months, and x pairs after (n+1) months. That does not say anything about how to get from x to x[n], however.

In month (n), how many pairs of rabbits are old enough to breed? All who were alive in the previous month, numbered (n-1). So in month (n), exactly x rabbits will be born. This number, “x”, will be added to the “x[n]” rabbits alive in month (n) to create the number x. To put it mathematically, x = x[n] + x.

So the sequence was important to Fibonacci because he won a contest with it. Later, it became important for other reasons.

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