Transcript Sequence - University of New Mexico

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?