The Fibonacci Sequence

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The Fibonacci Sequence

• By Steven Cornell

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• Was created by Leonardo Pisano Bogollo.
• It shows the growth of an idealized rabbit
  population.

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Rabbits

• They mate after one month
• After a month a new pair of rabbits are born
• The third month, the first pair creates another
  pair (so there are 3 pairs now).
• The fourth month the first 2 create new pairs (so
  5 exist now).
• Etc.

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Numbers

• In the sequence, the next number is found by
  adding up the preceding two numbers in the
  sequence or with the equation
• Xn Xn-1 Xn-2

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Example

• Find the fourth term of the sequence
• Xn Xn-1 Xn-2
• Xn 1 0 1
• Xn 1 1 2
• Xn 2 1 3
• Xn 3 2 5
• SO, the fourth term is 5.

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Golden Spiral

• First take the numbers of the series and form
  them into boxes with the sides equal to the
  numbers, then place them in an outward spiral
• When an arc is drawn through the boxes from one
  corner to the opposing one it forms a spiral
  known as the Golden Spiral which can be seen in a
  nautilus shell.

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Nautilus Shell
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The Golden Number

• When a ratio of the numbers in the Fibonacci
  series is made, it is seen that they approach the
  number that is approximately 1.618034 (which is
  also known as the golden number).
• 1/1 1, 2/1 2, 3/2 1.5, 5/3 1.6666,
• 8/5 1.6, 13/8 1.625, 21/13 1.6153846, etc.

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Graph Showing the Golden Number
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Why does this sequence matter?

• Its used very often in nature and can be seen in
  the arrangement of leaves on a stem, the
  branching of trees, the flowering of artichokes,
  etc.