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Transcript
```Sequences
defined recursively
A Sequence is a set of numbers, called terms,
arranged in a particular order.
If the relation between the number n and the nth term can be expressed
by a formula , we call this formula the formula the general term
Example
tn  4n  1892
(1) please find the first five terms of the sequence.
(2) please find the 29th term of the sequence.
Sometimes a sequence is defined by the given
the value of tn in terms of the preceding term.
Example
t1  3

t n  2t n 1  1
please find the first five terms of the sequence.
The formulas
t1  3

tn  2tn 1  1
give a recursive definition for the sequence 3, 7 ,15, 31, 63, …
A recursive definition consists of two parts:
1. An initial condition that tells where the sequence starts.
2. A recursition formula that tells how any term in the sequence
is related to the preceding term.
1. Arithmetic Sequences
A Sequence of numbers is called an arithmetic sequence
if the difference of any two consecutive terms is
constant. This difference is called the common
difference.
Recursive
1
definition
t

t n  t n1  d
Formula for the nth general term
sequence
of an Arithmetic
t n  t1  (n  1)d
2. Geometric Sequences
A Sequences of numbers is called a geometric sequence
if the ratio of any two consecutive terms is constant.
This ratio is called the common ratio.
Recursive
definition
t1

t n  t n 1  r
Formula for the nth general term of a Geometric sequence
tn  t1  r
(n1)
3. 1 Linear of Step Sequences of recurrence
The Tower of Hanoï puzzle consists of a block of wood with three
posts, A, B, and C. On post A there are 5 disks of diminishing size from
bottom to top. The task is to transfer all 5 disks from post A to one of
the other two posts given that:
1.only one disk can be moved at a time
2.no disk can be placed on top of a smaller disk.
a. Let Mn represent the minimum number of moves needed to
move n disks from post A to one of the other posts. What
are M1, M2 and M3. …?
b. Suppose you know how to move (n-1) disks from post A to
another post, and that to do so requires Mn-1 moves. Find
the relation between the Mn and Mn-1.
and M3.Then find M4, M5, M6, M7, M8.
d. Find the formula for the nth term
Mn = Mn-1 + 1 + Mn-1
Mn + 1 = 2Mn-1 + 2 = 2(Mn-1 + 1)
M n + 1 = 2 n  Mn = 2 n - 1
e. According to legend, in the great Temple of Benares,there is
an altar with three diamond needles.
At the beginning of time, 64 gold rings of decreasing radius
from bottom to top were placed on one of the needles.
Day and night ,priests sit before the altar transferring one
gold ring per second in accordance with the two rules given
given above.
The legend also say that when all 64 rings have been
transferred to one of the diamond needles, the word will
come to an end.
How long will it take the priests to transfer all the rings?
M64 = 264 – 1 seconds = 584,5 billion years
which is more than 43 times the age of the Universe
4. Fibonacci Sequence
Here is a famous problem posed in the
XIII th century by Leonardo de Pisano ,
better known as Fibonacci : suppose we
have one pair of newborn rabbits of both
genders. We assume that the following conditions are true :
1. It takes a newborn rabbit one month to become an adult.
2. A pair of adult rabbits of both genders will produce one
pair of newborn rabbits of both genders each month ,
beginning one month after becoming adults.
3. The rabbits do not die
How many rabbits will there be one year later?
solution
1st month 1 pair
solution
1st month 1 pair
2nd month 1 pair
solution
1st month 1 pair
2nd month 1 pair
3rd month 2pairs
solution
1st month 1 pair
2nd month 1 pair
3rd month 2 pairs
4th month 3 pairs
solution
1st month 1 pair
2nd month
3rd month
4th month
5th month
1 pair
2 pairs
3 pairs
5 pairs
solution
1st month 1 pair
2nd month
3rd month
4th month
5th month
6th month
1 pair
2 pairs
3 pairs
5 pairs
8 pairs
solution
1st month 1 pair
2nd month
3rd month
4th month
5th month
6th month
7th month
1 pair
2 pairs
3 pairs
5 pairs
8 pairs
13 pairs
solution
Recursion formula:
 F1  1,
F2  1

 Fn  Fn1  F n 2
1 st
2 nd
3 rd
4 th
5 th
6th
1
1
2
3
5
8
7 th
8 th
9 th
13
21
34
10 th 11 th 12 th
55
89
144
• There will be 144 pairs of rabbits
after a year, that ia 288 rabbits
running around.
Each number in the Fibonacci sequence is called a Fibonacci
number（Fibonacci 数）. Probably most of us have never taken the
time to examine very carefully the number or arrangements (排列)
of petals（花瓣 on a flower.
If we were to do so, several things would become apparent.
We would find that the number of petals on a flower is often one of
the Fibonacci numbers.
calla（1）
begonia （2）
Trillium（3）
Orchid（5）
Bloodroot (8)
Daisy (13)
There is a kind of plant here.
Can you find something interesting?
13
8
5
3
2
1
1
Now, we will draw a picture showing the Fibonacci numbers.
We start with 2 small squares of size 1 next to each other.
On top of these draw a square of size 2.Then draw a new square of
size 3 just as the picture shows, and the square of size 5, size 8, size
13.we can draw a spiral by putting together quarter circles, one in
each square.
This is a spiral(螺旋线), a similar curve to this occurs in nature
as the shape of a nail shell or some sea shells.
(Show the shell I picked on the seaside .)
Some similar curves appear in pine cones（松果）. There are
closewise spirals and couter-clockwise spirals on pine cones. If
you have a good study on pine cones , you can find that the
numbers of seeds on sprials are also Fibonacci numbers .
The formula of the general term of the
Fibonacci number sequence is






1
1 5
1 5


Fn 




5  2   2  


Binet’s Formula
n
(French, XVIIIth Century)
n
If we take the ratio（比例 of two successive numbers
in Fibonacci series and we divide each by the number
before it, we will get the following series of numbers.
The ratio seems to approach to a particular
number, which we call the golden number
(黄金数)
It is often represented by the Greek letter phi(φ).
It is also appearing in many places in Nature.
Chain of 9 rings
… a game from ancient china …
```