The Mathematics
of the Fibonacci series
Fibonacci Series Java Applet
|
To run the applet, click on the
text field at the bottom.
Enter the number of terms you want
and RETURN.
The series will appear in the main
window.
You cannot have more than 91 terms
This applet should work very well with  .
If you are using  ,
you might have to reload after you keyed in the value for the scroll bar
to be displayed. |
Patterns in the Fibonacci Numbers
Cycles in the Fibonacci numbers
Here are some patterns people have already noticed:
-
There is a cycle in the units column - the
cycle of units digits (0,1,1,2,3,5,8,3,1,4,...) repeats from n=60 and again
every 60 values.
-
There is also a cycle in the last two digits,
repeating (00, 01, 01, 02, 03, 05, 08, 13, ...) from n=300 with a cycle
of length 300.
-
For the last three digits, the cycle length
is 1,500
-
for the last four digits,the cycle length
is 15,000 and
-
for the last five digits the cycle length
is 150,000
-
and so on...
Factors of Fibonacci Numbers
Things to do
-
Where are the prime numbers in this list?
-
Where are the even numbers?
-
What about the multiples of 3?
-
..or 5?
From the previous Things to Do you will have
found that...
-
The even Fibonacci numbers are F(3), F(6),
F(9), F(12), ... ie F(3k) or
every third Fibonacci number is a multiple
of 2
Notice that 2=F(3) also.
-
Those Fibonacci numbers which are multiples of
3 are:
F(4), F(8), F(12), F(16), ... or F(4k) so
every fourth Fibonacci number is a multiple
of 3.
Again notice that 3=F(4).
This suggests the following:-
-
Every fifth Fibonacci number is a multiple
of F(5)=5,
-
every sixth Fibonacci number is a multiple
of F(6)=8
The general rule is therefore:
Every kth Fibonacci number
is a multiple of F(k)
or, expressed mathematically,
F(nk) is a multiple of F(k)
for all values of n and k=1,2,...
This means that if the subscript is composite
(not a prime) then so is that Fibonacci number (with one exception - can
you find it?) So we now deduce that
Any prime Fibonacci number
must have a subscript which is prime
(with one little exception - can you find
it?)
Unfortunately, the converse is not always true
(that is, it is not true that if a subscript is prime then so is that Fibonacci
number). The first case to show this is the 19th position (and 19 is prime)
but F(19)=4181 and F(19) is not prime as 4181=113x37.
Fibonacci's Rabbit Generations and Pascal's Triangle
Here's another explanation of how the Pascal triangle numbers sum to give
the Fibonacci numbers, this time explained in terms of our original rabbit
problem.
Let's return to Fibonacci's rabbit problem and look at it another way.
We shall be returning to it several more times yet in these pages - and
each time we will discover something different!
We shall make
a family tree of the rabbits but this time we shall be interested only
in the females and ignore any males in the population. If you like,
in the diagram of the rabbit pairs shown here, assume that the rabbit on
the left of each pair is male (say) and so the other is female. Now ignore
the rabbit on the left in each pair!
We will assume that each mating produces exactly one female
and perhaps some males too but we only show the females in the diagram
on the left. Also in the diagram on the left we see that each individual
rabbit appears several times. For instance, the original brown female was
mated with a while male and, since they never die, they both appear once
on every line.
Now, in our new family tree diagram, each female rabbit will appear
only once. As more rabbits are born, so the Family tree grows
adding a new entry for each newly born female.
As in an ordinary human family tree, we shall show parents above a line
of all their children.
Here is a ficitious human family tree with the names of the relatives
shown for a person marked as ME:
Grandpa Grandma Grandma Grandpa
Abel===Mabel Freda=====Fred
| |
| | Aunty Aunt Uncle
Uncle Bob---Dad=============Mum----Jane-----Hayley=Clement
| |
sister-in-law |brother sister |
Joan===John---ME---Jean Cousin--Cousin
| Sonny Gale===Gustof
nephew Dan--neice Pam
The diagram shows that:
Grandpa Abel and Grandma Mabel are the parents of my Dad and
Grandma Freda and Grandpa Fred are the parents of my Mum.
Bob is my Dad's brother and
my Mum has two sisters, my aunts Hayley and Jane.
Aunt Hayley became Hayley Weather when she married Clement Weather.
They have two children, my cousins Sonny Weather and Gale Weather.
Gale married Gustof Wind and so is now Gale Wind.
My brother John and his wife Joan have two children,
my nephew Dan and my neice Pam.
In this family tree of human relationships, the === joins people who
are parents or signifies a marriage.
In our rabbit's family tree, rabbits don't marry of course, so we just
have the vertical and horizontal lines:
-
The vertical line |
-
points from a mother (above) to the oldest daughter (below);
-
the horizontal line -
-
is drawn between sisters from the oldest on the left down to the youngest
on the right;
-
the small letter r
-
represents a young female and
-
the large letter R
-
shows a mature female who can and does mate every month, producing a new
daughter each month.
As in Fibonacci's original problem (in its variant form that makes it a
bit more realistic) we assume none die and that each month every mature
female rabbit always produces a babies of which exactly one is a female.
Here is the Rabbit Family tree as if grows month by month for the first
8 months:
M o n t h
1 2 3 4 5 6 7 8
r R R R R R R R
| | | | | |
r R_r R_R_r R___R_R_r R_____R___R_R_r R_________R_____R___R_R_r
| | | | | | | | |
r R_r r R_R_r R_r r R___R_R_r R_R_r R_r r
| | | |
r R_r r r
So in month 2, our young female (r of month 1) becomes mature (R)
and mates.
In month 3, she becomes a parent for the first time and produces
her first daughter, shown on a line below - a new generation.
In month 4, the female born in month 2 becomes mature (R) and
also her mother produces another daughter (r).
In month 5, the original female produces another female child
added to the end of the line of the generation of her daughters, while
the daughter born the previous month (the second in the line) becomes mature.
Also the first daughter produces her own first daughter, so in month 5
the original female becomes a grand-mother and we have started a third
line - the third generation.
The Family tree is shown for the first 8 months as more females are
added to it. We can see that our original female becomes a great-grandmother
in month 7 when a fourth line is added to the Family tree diagram - a fourth
generation!
Have you spotted the Pascal's triangle numbers in the Rabbit's
Family Tree?
The numbers of rabbits in each generation, that is, along each level (line)
of the tree, are the Pascal's triangle numbers that add up to give each
Fibonacci number - the total number of (female) rabbits in the Tree. In
month n there are a total of F(n) rabbits, a number made up from the entry
in row (n-k) and column (k-1) of Pascal's triangle for each of the levels
(generations) k from 1 to n. In other words, we are looking at this formula
and explaining it in terms of generations, the original rabbit forming
generation 1 and her daughters being generation 2 and so on:
n
----- / \
\ | n - k |
Fib(n) = ) | |
/ | k - 1 |
----- \ /
k = 1
Remember that the rows and columns of Pascal's triangle in this formula
begin at 0!
For example, in month 8, there are 4 levels and the number on each
level is:
M o n t h 8:
Level 1: 1 rabbit which is Pascal's triangle row 7=8-1 and column 0=1-1
Level 2: 6 rabbits which is Pascal's triangle row 6=8-2 and column 1=2-1
Level 3: 10 rabbits which is Pascal's triangle row 5=8-3 and column 2=3-1
Level 4: 4 rabbit which is Pascal's triangle row 4=8-4 and column 3=4-1
When k is bigger than 4, the column number exceeds the row number in Pascal's
Triangle and all those entries are 0.
SUM is F(8)=21
col : 0 1 2 3 4 5 6 7 8 /9 ...
------+--------------------------/-----
0 | 1 0 0 0 0 0 0 0 0...
r 1 | 1 1 0 0 0 0 0 0 0...
o 2 | 1 2 1 0 0 0 0 0 0...
w 3 | 1 3 3 1 0 0 0 0 0...
4 | 1 4 6 4 1 0 0 0 0...
5 | 1 5 10 10 5 1 0 0 0...
6 | 1 6 15 20 15 6 1 0 0...
7 | 1 7 21 35 35 21 7 1 0...
8 | 1 8 28 56 70 56 28 8 1...
... ...
The general pattern for month n and level (generation) k is
Level k: is Pascal's triangle row n-k and column k-1 For month n we
sum all the generations as k goes from 1 to n (but half of these will be
zeros).
Things to do
-
Make a diagram
of your own family tree. How far back can you go? You will probably have
to ask your relatives to fill in the parts of the tree that you don't know,
so take your tree with you on family visits and keep extending it as you
learn about your ancestors!
-
Start again and
draw the Female Rabbit Family tree, extending it month by month. Don't
distinguish between r and R on the tree, but draw the newly born rabbits
using a new colour for each month or, instead of using lots of colours,
you could just put a number by each rabbit showing in which month it was
born.
-
If you tossed
a coin 10 times, how many possible sequences of Heads and Tails could there
be in total (use Pascal's Triangle extending it to the row numbered 10)?
In how many
of these are there 5 heads (and so 5 tails)? What is the probability of
tossing 10 coins and getting exactly 5 heads therefor - it is not
0.5! Draw up a table for each even number of coins from 2 to 10
and show the probability of getting exactly half heads and half tails for
each case. What is happening to the probablilty as the number of coins
gets larger?
-
Draw a histogram
of the 10th row of Pascal's triangle, that is, a bar chart,
where each column on the row numbered 10 is hown as a bar whose height
is the Pascal's triangle number. Try it again for tow 20 if you can (or
use a Spreadsheet on your computer). The shape that you get as the row
increases is called a Bell curve since it looks like a bell cut
in half. It has many uses in Statistics and is a very important
shape.
-
Make a Galton
Quincunx.
This is a
device with lots of nails put in a regular hexagon arrangement (its name
derives from the Latin word quincunx for the X-like shape the 5
spots form on the 5-face of a dice.)
\ ooo / Galton's Quincunx
\ooooo/
\ooo/ funnel to direct the balls
\o/ directly on to
/ . \ the topmost nail
/ . . \
/ . . . \
/ . . . . \ rows
/ . . . . . \ of
/ . . . . . . \ nails
/ . . . . . . . \
/ . . . . . . . . \
| | | | | | | | | |
| | | | |o| | | | | Containers to collect the
| | | |o|o|o|o| | | balls as they fall through
|o| |o|o|o|o|o|o| |
-------------------
The whole board is tilted slightly so that the
top is raised off the table a little. When small balls are poured onto
the network of nails at the top, they fall through, bouncing either to
the right or to the left and so hit another nail on the row below. Eventually
they fall off the bottom row of nails and are caught in containers.
If you have a lot of nails and a lot of little
balls (good sources for these are lead shot from a fishing shop or small
steel ball-bearings from a cycle shop or even dried peas or other cheap
round seeds from the supermarket) then they end up forming a shape in the
containers that is very much like the Bell curve of the previous exploration.
[You could try simulating this experiment
on a computer using its random number generator to decide which side the
ball bounces on (say the random number is between 0 and 1 then, if it is
between 0 and 0.5 it goes to the left and above 0.5 meamsn it bounces to
the right. Do this at least 10 times, keeping a track of which nail along
the row it bonces on to so that, at the last row, you know which container
it will fall in to.]
Let's see how the curve of the last two explorations,
the Bell curve might actually occur in some real data sets.
Measure the height of each person in your
class and plot a graph similar to the containers above, labelled with heights
to the nearest centimetre, each container containing one ball for each
person with that height. What shape do you get? Try adding in the results
from other classes to get one big graph.
This makes a good practical demonstration
for a Science Fair or Parents' Exhibition or Open Day at your school or
college. Measure the height of each person who passes your display
and "add a ball" to the container which represents their height. What shape
do you get at the end of the day?
What else could you measure?
-
The weight of each person to the nearest ounce
or 10 grams;
-
their age last birthday;
-
their house or apartment number;
-
the last 3 digits of their telephone number;
or try these data sets using coins and dice:
-
the total number when you add the spots after
throwing 5 dice at once;
-
the number of heads when you toss 20 coins at
once.
Do all of these give the Bell curve for
large samples?
If not, why do you think some do and some don't?
Can you decide beforehand which will give
the Bell curve and which won't?
Write out the first few powers of 11. Do they
remind you of Pascal's triangle? Why? Why does the Pascal's triangle pattern
break down after a few powers?
(Hint: consider (a+b)m where a=10
and b=1).
To finish, let's return to a human family
tree. Suppose that the probability of each child being male is exactly
0.5. So a new baby will be male half the time and female the other half.
If a couple have 2 children, what are the four possible sequences of children
they can have? What is it if they have 3 children? Suppose a couple have
5 children. In what proportion of the couples that have 5 children will
all 5 children be boys?
