The Fibonacci Sequence are a series of numbers created by adding together the two preceding numbers. This seemingly innocuous sequence crops up in mathematics and the natural sciences with bizarre regularity. The sequence had been known to Indian mathematicians since at least 200 BC, but it was first bought to the West by 13th century Italian mathematician Leonardo of Pisa. Later renamed Leonardo Fibonacci (to distinguish him from another famous Pisan Leonardo), the mathematician wrote about this mysterious sequence in his book Liber Abaci. He used it to describe the breeding patterns of rabbits, but that’s barely scratching the surface. Let’s learn about this strange sequence of numbers that appears in everything from biology to finance.
By analyzing questions, you can see patterns emerge, patterns that will help you answer questions. Qwiz5 is all about those patterns. In each installment of Qwiz5, we take an answer line and look at its five most common clues. Here we explore five clues that will help you answer a tossup on the Fibonacci Sequence.
THE GOLDEN RATIO
As the numbers in the Fibonacci Sequence increase towards infinity, the ratio of two adjacent terms approaches something called the Golden Ratio. This ratio is an irrational number created when a line is divided into two parts such that the long part divided by the short part equals the total length divided by the long part. Shapes whose sides conform to the Golden Ratio are thought to be visually pleasing, hence its recurrence in many works of art and architecture.
Binet’s Formula is an explicit formula utilized to find the nth term in the Fibonacci Sequence. Binet’s Formula is called a closed form formula for the Fibonacci Sequence, because it can be used to calculate any number in the sequence without having to know the numbers immediately before or after it.
The Lucas Numbers, or Lucas Series, are a sequence of numbers similar to the Fibonacci Sequence. Named after French mathematician François Lucas, the sequence is calculated by adding together the two preceding terms. Unlike the Fibonacci Sequence, however, the Lucas sequence begins with the input 2, 1. The Fibonacci Sequence can be thought of as a specific form of Lucas numbers.
Given two natural numbers a and b, Euclid’s algorithm finds their greatest common divisor. Euclid’s Algorithm has a bit of a fit when dealing with two consecutive Fibonacci numbers, however. The algorithm requires many more steps than usual to find the greatest common divisor of two neighboring Fibonacci numbers. This is called a worst-case runtime.
Zeckendorf’s Theorem displays another interesting feature of the Fibonacci numbers. The Theorem states that every positive integer can be written as the sum of two non-neighboring Fibonacci numbers. Non-neighboring numbers are two numbers in the sequence that are not immediately next to each other.
Quizbowl is about learning, not rote memorization, so we encourage you to use this as a springboard for further reading rather than as an endpoint. Here are a few things to check out:
* The Fibonacci Sequence may not be the “secret code” of the universe, but its numbers describe many natural phenomena!
* The Fibonacci Sequence appears in nature, but it also appears in Wall Street.
* There are many strange properties of the Fibonacci Sequence for the more mathematically inclined to study!
* We talk about the Golden Ratio in visual arts all the time. What if we were to apply the Fibonacci Sequence to the musical arts?
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