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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 n1 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 (n1) 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. c. Use your answer to part (b) to check your values for M2 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 Fn1 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. 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 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 …