Mathematics: Recursive Sequence

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• A sequence where a new term in defined from the previous terms is called recursive.
  • 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ... is recursive because each new term is the sum of the previous two.=1
• The recursive formula in this example is f₁=1, f₂=1, f_n = f_n-1 + f_n-2
• This apparently simple recursive formula is one of the best known examples and is known as the Fibonacci sequence.
• This sequence is the answer to the question posed by Fibonacci in 1202
  • How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?
  • One pair will be produced the first month, and 1 pair in the second month. because the new pair produced in the first month is not yet mature.  In the third month 2 pairs will be produced, one by the original pair and one by the pair which was produced in the first month. In the fourth month 3 pairs will be produced, and in the fifth month 5 pairs. After this things expand rapidly. This simple, seemingly unremarkable recursive sequence has fascinated mathematicians for centuries.
  • Consider the following diagram made of square whose lengths correspond to the Fibonacci sequence and are organized in an outwardly spiraling pattern.

Notice that the rectangles formed at each stage are roughly the same shape. The ancient Greeks considered the proportion of this rectangle to be the aesthetically perfect proportion.  The Greeks used this proportion in much of their artwork and architecture.  A rectangle whose sides have this proportion was called the Golden Rectangle. The number that represents the ratio of the two sides of the rectangle is known as the Golden Mean and is usually denoted with the Greek letter phi Ф.

1 + √5  = Ф = The Golden Mean
    2

To prove this, consider the sequence of rectangle side ratios for the spiraling squares;

This sequence converges to the Golden Mean number.

The general form of the sequence of ratios is;

t₁ = 1/1, t₂ = 2/1, t_n = f_n+1/f_n where f_n is a term in the Fibonacci sequence. 

Then t_n = f_n+1/f_n = (f_n+f_n-1)/f_n = 1 + f_n-1/f_n  = 1 + 1/(f_n/f_n-1) = 1 + 1/t_n-1 

We will not prove it here but accept that the sequence converges to a real number t, then

lim t_n = lim t_n-1 = t
^{n―›∞     n―›∞}

We then have t = 1 + 1/t which is a quadratic equation which we can solve and show that

1 + √5  = t = Ф = The Golden Mean
    2

Honeybee Example

Female honeybees hatch from an egg that has been fertilized by a male honeybee, but male honeybees hatch from unfertilized eggs and have only one female parent. The ancestral tree of a male honeybee shows the Fibonacci sequence.