Art and Nature: The Fibonacci Sequence
Leonardo Pisano Fibonacci was a famous, Italian mathematician known for his Fibonacci
sequence. The original problem that Fibonacci investigated (in the year 1202) was about how
fast rabbits could breed in ideal circumstances.
His project: Suppose a newly-born pair of rabbits, one male, one female, are put in a
field. Rabbits are able to mate at the age of one month so that at the end of its second month a
female can produce another pair of rabbits. Suppose that our rabbits never die and that the
female always produces one new pair (one male, one female) every month from the second
month on. The puzzle that Fibonacci posed was...How many pairs will there be in one year?
The sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... This sequence shows that each number is
the sum of the two preceding numbers. It is a sequence that is seen and used in many different
areas of mathematics and science today.
We know that the first 10 Fibonacci numbers are {1, 1, 2, 3, 5, 8, 13, 21, 34, 55}. We use the
notation 𝐹𝑛 to represent the nth Fibonacci number – that is, 𝐹1 = 1, 𝐹2 = 1, 𝐹3 = 2, etc. Also
remember that we find the nth Fibonacci number by adding together the two that come before
it. Find the numerical value of the following:
(a)
𝐹11
(b)
𝐹12
(c)
𝐹12+1
(d)
𝐹12 + 1
Given that 𝐹36 = 14, 930, 352 and 𝐹37 = 24, 157, 817, find:
(a)
𝐹38
(a)
𝐹35
Ex. The following sequence is a Fibonacci-type sequence. Find the missing terms.
7, ___, ___, 27, 44, ___
Let a represent the 1000th Fibonacci number and b represent the 1001 st Fibonacci number.
Express the 1003rd Fibonacci number in terms of a and b. (In other words, you’re doing this
without ever knowing what the 1000th and 1001st Fibonacci numbers are.) Simplify your
answer.