Satyagopal Mandal
Department of Mathematics
University of Kansas
Office: 624 Snow Hall  Phone: 785-864-5180
  • e-mail: mandal@math.ukans.edu
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    Chapter 9

    Fibonacci Numbers and the Golden Ratio

     

    Recall that a list of numbers

    a1, a2, a3, …, an, …..

    is called a sequence of numbers. So, the first term of ths sequence is a1, the second term of the sequence is a2, and so on.

     

    Definition: The sequence

    1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …..

    is called the Fibinacci sequence and these numbers are called the Fibinacci numbers. So,

    1. The first term of the sequence is F1 = 1

    2. The 2nd term of the sequence is F2 = 1

    3. The 3rd term of the sequence is F3 = 2

    4. The 4th term of the sequence is F4 = 3

    5. The 5th term of the sequence is F5 = 5

    6. The 6th term of the sequence is F6 = 8

    7. The 7th term of the sequence is F7 = 13

    8. The 8th term of the sequence is F8 = 21

    9. The 9th term of the sequence is F9 = 34

    10. The 10th term of the sequence is F10 = 55

    11. The 11th term of the sequence is F11 = 89

    12. The 12th term of the sequence is F4 = 144

    ……

    N) In general, FN denotes the Nth-term of the sequence.

     

    What is the value of FN? To answer that you have to understand the pattern of the sequence. Note that

    1. 2 = F3 =1 + 1 = F1+ F2

    2. 3 = F4 =1 + 2 = F2+ F3

    3. 5 = F5 =2 + 3 = F3+ F4

    4. 8 = F6 =3 + 5 = F4+ F5

    5. 13 = F7 =5 + 8 = F5+ F6

    6. 21 = F8 = 8 + 13 = F6+ F7

    7. 34 = F9 =13 + 21 = F7+ F8

    8. 55 = F10 = 21 + 34 = F8+ F9

    9. 89 = F11 =34 + 55 = F9+ F10

    10. 144 = F12 =55 + 89 = F10+ F11

     

    So, we observe that, except for F1 and F2, each term is the sum of the two preceding terms. So, we have

    F1 =1 F2 = 1 and FN = FN-2+ FN-1 (for N>2).

     

    Suggested Problems: Ex. 1,2,3,5,6 (pp. 318).

     

    The Formula: The Fibonacci numbers can be directly computed by the Binet's Formula:

     

    FN= (((1 + 5 1/2)/2)N + ((1 - 5 1/2)/2)N)/ 5 1/2

     

    Definition: The number f = (1 + 5 1/2)/2 is called the golden ratio. According to the calculator, approximately, (correct up to a certain decimal point)

    f = 1.618033989.

     

    Let us also define another number t = (1 - 5 1/2)/2. According to the calculator, approximately, (correct up to a certain decimal point)

    t = - 0.6180339887.

     

    Remark 1: Note that these two numbers f, t are solutions of the quadratic equation x2 = x+1.

     

    Remark 2: One can also check that fN = FNf+FN-1.

     

    Remark 3: This golden number f and the Fibonacci numbers appear in nature and geometry, art, architecture. Your textbook went through a very interesting discussion on this. It will be an interesting reading for you.

     

    Some Geometry

    Definition: A gnomon to a figure A, is a connected figure B, which, when suitably attached to A produces a new figure which is similar to A. If "G&A" is similar to A then G is a gnomon to A.

     

    Recall, in geometry, we say two figures are similar if one is a scaled version of the other. We will have further discussion on gnomons in the class. This part will not be available on the web.

     

    Suggested Problems: Examples 1-10 Ex. 28-32 (pp. 321).