sequences defined recursively. a sequence is a set of numbers, called terms, arranged in a...

Sequences defined recursively

A Sequence is a set of numbers, called terms, arranged in a paticurlar order.

Example

18924 ntn

(1) please find the first five terms of the sequence.

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

(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

12

3

1

1

nn tt

t

please find the first five terms of the sequence.

12

3

1

1

nn tt

tThe formulas

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.

dnttn )1(1

Formula for the nth term of general arithmetic sequence

dtt

t

nn 1

1Recursive definition

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.

)1(1

nn rtt

Formula for the nth term of general geometric sequence

Recursive definition

rtt

t

nn 1

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?

solution1st month 1 pair

solution1st month 1 pair

2nd month 1 pair

solution1st month 1 pair

2nd month 1 pair3rd month 2pairs

solution1st month 1 pair

2nd month 1 pair3rd month 2 pairs

4th month 3 pairs

solution1st month 1 pair

2nd month 1 pair3rd month 2 pairs4th month 3 pairs5th month 5 pairs

solution

6th month 8 pairs

1st month 1 pair

2nd month 1 pair

3rd month 2 pairs4th month 3 pairs5th month 5 pairs

solution

6th month 8 pairs

1st month 1 pair

2nd month 1 pair

3rd month 2 pairs4th month 3 pairs5th month 5 pairs

7th month 13 pairs

solution

1 st 2 nd 3 rd 5 th4 th 6th

7 th 8 th 9 th 11 th10 th 12 th

1 1 2 3 5 8

13 21 34 55 89 144

• There will be 144 pairs of rabbits after a year.

recursition formula:

21

21 1,1

nnn FFF

FF

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.

begonia （ 2）calla （ 1） Trillium （ 3 ）

orchid（ 5）

daisy(13) bloodroot(8)

138

53211

There is a kind of plant here. Please observe it carefully. Can you find something interesting?

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