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Fibonacci’s rabbits

Fibonacci’s rabbits. Fibonacci posed the following problem: A mature pair of rabbits can give birth to a new pair of baby rabbits every month. Juvenile rabbits require 1 month to mature, and then they too start breeding.

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Fibonacci’s rabbits

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  1. Fibonacci’s rabbits Fibonacci posed the following problem: A mature pair of rabbits can give birth to a new pair of baby rabbits every month. Juvenile rabbits require 1 month to mature, and then they too start breeding. If you start with 1 pair of juvenile rabbits, how many pairs of rabbits will you have after 3 months? 6 months? 1 year? Is there a pattern that will let us determine how many rabbits we will have after N months, for any N ?

  2. Initially: 1 pair of juvenile rabbits

  3. After 1 month The pair of rabbits are now mature

  4. After 2 months The original pair of rabbits are (still) mature, and there is now a new pair of juvenile offspring.

  5. After 3 months….

  6. And 4...

  7. And 5...

  8. And 6...

  9. And 7...

  10. During this there are this many and this many pairs total number month pairs of young rabbits of mature rabbits

  11. During this there are this many and this many pairs total number month pairs of young rabbits of mature rabbits 1 11

  12. During this there are this many and this many pairs total number month pairs of young rabbits of mature rabbits 1 11 2 11 Any rabbits that were around initially are mature after 1 month.

  13. During this there are this many and this many pairs total number month pairs of young rabbits of mature rabbits 1 11 2 11 3 112 Any rabbits that were around after 1 month are mature after 2 months. Any rabbits that were mature after 1 month give birth to offspring in the 2nd month.

  14. During this there are this many and this many pairs total number month pairs of young rabbits of mature rabbits 1 11 2 11 3 112 4 123 Any rabbits that were around after 2 months are mature after 3 months. Any rabbits that were mature after 2 month give birth to offspring in the 3rd month.

  15. During this there are this many and this many pairs total number month pairs of young rabbits of mature rabbits 1 11 2 11 3 112 4 123 5 235

  16. During this there are this many and this many pairs total number month pairs of young rabbits of mature rabbits 1 11 2 11 3 112 4 123 5 235 6 358

  17. During this there are this many and this many pairs total number month pairs of young rabbits of mature rabbits 1 11 2 11 3 112 4 123 5 235 6 358 7 5813

  18. During this there are this many and this many pairs total number month pairs of young rabbits of mature rabbits 1 11 2 11 3 112 4 123 5 235 6 358 7 5813 8 81321

  19. General Pattern: In any given month, the total number of rabbits = number of mature rabbits + number of juvenile rabbits.

  20. General Pattern: In any given month, the total number of rabbits = number of mature rabbits + number of juvenile rabbits. But thenumber of mature rabbits =the total number of rabbits from the previous month, and the number of juvenile rabbits = the number of mature rabbits from the previous month, which is the same as the total number of rabbits from 2 months prior.

  21. General Pattern: In any given month, the total number of rabbits = number of mature rabbits + number of juvenile rabbits. But the number of mature rabbits = the total number of rabbits from the previous month, and the number of juvenile rabbits = the number of mature rabbits from the previous month, which is the same as the total number of rabbits from 2 months prior. Conclusion: The number of rabbits in any month can be found by adding the number of rabbits one month earlier to the number of rabbits 2 months earlier.

  22. General Pattern: Let fn represent the total number of rabbits after n months. Then, we have the formula fn = fn-1 + fn-2 number of rabbits after number after + number after n months n-1 months n-2 months

  23. After this many months… there are this many pairs of rabbits 1 1 2 1 3 2 4 3 5 5 6 8 7 13 8 21 General formula… fn = fn-1 + fn-2

  24. After this many months… there are this many pairs of rabbits 1 1 2 1 3 2 4 3 5 5 6 8 7 13 8 21 General formula… fn = fn-1 + fn-2 with initial conditions f1 = f2 =1.

  25. The Fibonacci sequence: • 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…

  26. The Fibonacci sequence: • 1, 1, 2, 3, 5, 8, 13, 21, 34, 55… What is the next Fibonacci number?

  27. The Fibonacci sequence: • 1, 1, 2, 3, 5, 8, 13, 21, 34, 55… 34+55=89, so 89 comes next.

  28. The Fibonacci sequence: • 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … We now know how many rabbits there will be after 11 months – 1 more month and we have an answer to Fibonacci’s original question.

  29. The Fibonacci sequence: • 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … 55+89=144, so there are 144 pairs of rabbits after 1 year.

  30. A sequence is recursively defined if, as in Fibonacci’s example, the rule for finding the next number is based on the numbers already found.

  31. Another example of a recursively defined sequence might be: 10, 11, 12, 13… What comes next? What is the recursive rule?

  32. Recursively defined sequences are often easy to work with, but they do suffer from one serious drawback:

  33. Recursively defined sequences are often easy to work with, but they do suffer from one serious drawback: What is the 50th Fibonacci number?

  34. An alternative way to define a sequence is to explicitly define the terms – give an explicit formula, or algebraic rule, which tells you how to compute the nthnumber directly.

  35. Consider the sequence defined by the rule an=n2. What is the 1st term, a1? What is the 2nd term, a2? What is the 50th term?

  36. Let’s return for a moment to the sequence 10, 11, 12, 13, … What is an explicit formula for this sequence?

  37. 10, 11, 12, 13, … Since the first number is 10, we know we want a1=10

  38. 10, 11, 12, 13, … Since the first number is 10, we know we want a1=10 Also, we want our formula to cause an to increase by 1 every time n is increased by 1.

  39. 10, 11, 12, 13, … an=9+n meets both these requirements, so this is our explicit formula for this sequence.

  40. As with the last example, there is an explicit formula for the Fibonacci numbers, too. The problem is that it is a bit messy:

  41. The Fibonacci sequence, explicit formula: This is known as Binet’s formula:

  42. The number is important enough that it is given a symbol, Φ, the Greek letter Phi. A large portion of this chapter will deal with properties of Φ.

  43. By, the way, using Binet’s formula, replacing “N” with 50, we get = 12,586,269,025 which is the 50th Fibonacci number

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