Sequence - University of New Mexico
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Sequence
CHAPTER 6 SECTION 6
Definition
What is a sequence?
Sequence is a list of
numbers that follows some
rule or pattern.
Types of Sequences
1. Arithmetic Sequence- is a
sequence in which each term after
the first term differs from the
preceding term by a fixed constant.
Examples;
3, 7, 11, 15, …
-6, -2, 2, 6, …
Geometric sequence- is a sequence in
which each term, after the first
term is a nonzero constant multiple
of the preceding term. The constant
by which we are multiplying is called
the common ratio.
Examples;
3, -6, 12, -24, 48,…
2, 1, ½, ¼, 1/16, …
Identify the sequence as
either arithmetic or
geometric. List the next two
terms.
Problems
11, 7, 3, -1, …
1, ½, ¼, 1/8, …
1, -1, 1, -1,…
10, 5, 0 , -5, …
Solutions
Arithmetic; -5, -9
Geometric; 1/16, 1/32
Geometric; 1, -1
Arithmetic; -10, -15
Famous Sequences
Fibonacci
Fibonacci
was also known as Leonardo of Pisa
(1202).
Wrote a book, Liber Abaci, in which he
introduced the Hindu-Arabic numeration
system to Europe.
Introduced the most famous sequence in the
history of mathematics; The Fibonacci
Sequence.
What is the Fibonacci Sequence?
It is a sequence that was well known in ancient
India (200BC), before Fibonacci introduced it to
the west.
He considers the growth of an idealized
(biologically unrealistic) rabbit population,
assuming that: 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; rabbits never die and a
mating pair 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?
At the end of the first month, they mate, but there
is still one only 1 pair.
At the end of the second month the female produces
a new pair, so now there are 2 pairs of rabbits in the
field.
At the end of the third month, the original female
produces a second pair, making 3 pairs in all in the
field.
At the end of the fourth month, the original female
has produced yet another new pair, the female born
two months ago produces her first pair also, making 5
pairs.
Solution
In one year you will have 233
pairs.
Fibonacci Sequence
1, 1, 2, 3, 5, 8, 13, 21, 34,
55, 89 ,……….
Where does it occur?
This sequence occurs in nature in many
different ways.
Examples;
The seeds of daisies and sunflowers, are
arranged in spirals going in two
different directions.
Hexagonal “Bumps” on the skin of
pineapple.
Golden Ratio
What is the Golden
Ratio?
How is it related to
Fibonacci sequence?
I have answers
Known as φ (phi).
By dividing each term by
the term preceding it.
(See next slide)
1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.66
8/5 = 1.6
13/8 = 1.625
21/13 = 1.615
34/21 = 1.619
55/34 = 1.617
89/55 = 1.618
144/89 = 1.618
233/144 = 1.618
Squares
Why is this useful?
Architects still use this today
United Nations building.
Leonardo Di Vinci may have
used it in his paintings.
Homework
rectangle
length width ratio of length to width
Your cell phone
3" x 5" index card
8.5" X 11" paper
Your Head
Laptop
TV
Microwave
Stuff around
One more?