Download Recursive sequences By Wu laoshi in Paris in January 2009

Sequences
defined recursively
A Sequence is a set of numbers, called terms,
arranged in a paticurlar order.
If the relation between the number n and the nth term can be
expressed by a formula , we call this formula the formula of
general term
Example
t n  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 definited by givien 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

t n  2t n 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
definition
t1

t n  t n 1  d
Formula for the nth term of general arithmetic sequence
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 term of general geometric sequence
t n  t1  r
( n 1)
3. 1 linear of step Sequences of recurrence
The Tower of Hanio puzzle consists of a block of
wood with three posts, A, B, and C.On post A there
are eight disks of diminishing size from bottom to
top. The task is to transfer all eight 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.
c. Use your answer to part(b) to check you values for M2
and M3.Then find M4, M5, M6, M7, M8.
d. Find formula of the general term (formular for the nth
term)
e. According to legend, in the great Temple of Benarses,there
is an altar with three diamond needles, At the beginning of
time, 64 gold rings of decreasing radius from botom 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?
4. Fibonacci Sequence
Here is a famous problem posed in the
thirteenth centrry 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 wiil 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
recursition formula:
F2  1
F1  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.
Each number in the Fibonacci sequence is called Fibonacci number
（Fibonacci 数）. Probably most of us have never taken the time to
examine very carefully the number or arrangement(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. Please observe it
carefully. 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 two 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 general term of Fibonacci number sequence is
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 a Greek letter phi(φ).It is also a very
useful thing in our life and in the nature.
9 chain of ringses
----a game from ancient china