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Fibonacci sequences

Leonardo Fibonacci was a famous thirteenth century mathematician who discovered some very interesting patterns of numbers in nature.

Fibonacci’s rabbits

These rules determine how fast rabbits can breed in ideal circumstances.

• Generation 1: One pair of newborn rabbits is in a paddock. A pair is one female and one male.

• Generation 2: After it is 2 months old (in the third month), the female produces another pair of rabbits.

• Generation 3: After it is 3 months old (in the fourth month), this same female produces another pair of rabbits.

• Every female rabbit produces one new pair every month from age 2 months.

a Using the ‘rabbit breeding rules’, complete a drawing of the first five generations of rabbit pairs. Use it to complete the table below.

b Write down the numbers of pairs of rabbits at the end of each month for 12 months. This is the Fibonacci sequence.

c How many rabbits will there be after 1 year?

d Explain the rule for the Fibonacci sequence.

Fibonacci sequence in plants

a Count the clockwise and anticlockwise spiralling ‘lumps’ of some pineapples and show how these numbers relate to the Fibonacci sequence.

b Count the clockwise and anticlockwise spiralling scales of some pine cones and show how these numbers relate to the Fibonacci sequence.

Fibonacci sequence and the golden ratio

a Copy this table and extend it to show the next 10 terms of the Fibonacci sequence: b Write down a new set of numbers that is one Fibonacci number divided by its previous Fibonacci number. Copy and complete this table. c What do you notice about the new sequence (ratio)? d the golden ratio and explain how it links to your new sequence.

Problems and challenges

1 Sticks are arranged by a student such that the first three diagrams in the pattern are:

Up for a challenge? If you get stuck on a question, check out the ‘working with unfamiliar problems’ poster at the end of the book to help.

How many Sticks would there be in the 50th diagram of the pattern?

2 A number is said to be a ‘perfect number’ if the sum of its factors equals the number. For this exercise, we must exclude the number itself as one of the factors.

The number 6 is the first perfect number. Factors of 6 (excluding the numeral 6) are 1, 2 and 3 a Find the next perfect number. (Hint: It is less than 50.) b The third perfect number is 496. Find all the factors for this number and show that it is a perfect number. a What is the largest number of tulips Anya can put in each bunch? b How many bunches of each colour would Anya make with this number in each bunch?

The sum of these three factors is 1 + 2 + 3 = 6. Hence, we have a perfect number.

3 Anya is a florist who is making up bunches of tulips with every bunch having the same number of tulips. Anya uses only one colour in each bunch. She has 126 red tulips, 108 pink tulips and 144 yellow tulips. Anya wants to use all the tulips.

4 Mr and Mrs Adams have two teenage children. If the teenagers’ ages multiply together to give 252, find the sum of their ages.

5 Complete this sequence.

5 42 = 32 + 7 52 = 62 =

6 Use the digits 1, 9, 7 and 2, in any order, and any operations and brackets you like, to make as your answers the whole numbers 0 to 10. For example:

1 × 9 7 2 = 0

(9 7) ÷ 2 1 = 0

7 The first three shapes in a pattern made with sticks are:

How many sticks make up the 100th shape?

8 Two numbers have a highest common factor of 1. If one of the numbers is 36 and the other number is a positive integer less than 36, find all possible values for the number that is less than 36.