Fibonacci (meaning son of Bonaccio) was born in 1170 AD, and was the greatest Western mathematician during the Middle Ages. His full name was Leonardo of Pisa and he is famous for posing the following problem in his book *Liber Abaci* in 1202:

"A pair of newly born rabbits is brought into a confined place. This pair, and every later pair, begets one new pair every month, starting in their second month of age. How many pairs will there be after one, two, three, ... months, etc. assuming that no deaths occur?"As is well-known the solution to this problem gives us the famous Fibonacci sequence 1; 1; 2; 3; 5; 8; ... The Fibonacci sequence has many properties and is connected to many parts in mathematics, science, nature & art, and has even been used in novels such as the

**Explore**

In the interactive sketch below the first twenty terms *T*_{n} of the Fibonacci sequence were produced with the rule *T*_{n} + *T*_{n + 1} = *T*_{n + 2}, and are given in the 3rd column.

1) What do you notice about the ratio of consecutive numbers given in the last column?

2) Carefully compare the sum to *n* terms, *S _{n}*, in the 5th column with the terms

3) Click in the '

4) If necessary, repeat step 3), then formulate your observations in 1) and 2) above.

(Click on the '

A Fibonacci Generalization - Kendal's theorem

**Acknowledgement**

I'm greatly indebted to Scott Steketee for his valuable assistance in creating the *WebSketchpad* iterations shown above. For more detailed information on how to create & use iterations with Geometer's Sketchpad, Scott's webinar Exploring Mathematical Iteration is highly recommended.

**Explore More**

a) Click on the '**Link to k = 2**' button to navigate to a new sketch, which shows a new sequence & series created with the rule

5) What do you notice about the ratio of consecutive numbers given in the last column? Compare with 1) above: is it the same or different?

6) Carefully compare the sum to

7) Click in the '

8) If necessary, repeat step 7), then formulate your observations in 5) and 6) above.

(Click on the '

b) Click on the '

9) What do you notice about the ratio of consecutive numbers given in the last column? Compare with 1) & 5 above: is it the same or different?

10) Carefully compare the sum to

11) Note that for this page it is unfortunately not possible to edit the starting value or seed

12) Formulate your observations in 9) and 10) above.

(Click on the '

**Generalization**

13) Can you now generalize the Fibonacci rule *T*_{n} + *T*_{n + 1} = *T*_{n + 2} to a general recurrence rule in terms of *k*?

14) Also formulate conjectures regarding this general recurrence rule regarding your earlier observations about the ratio of consecutive terms as well as the relationship between *S _{n}* and

15) What about a converse of your conjecture in 14) regarding the relationship between

**Challenge **

16) Can you explain why (prove that) your observations & conjectures in 4), 8), 12), 14) and 15) above are true?

**Precious Metal Ratios**

For the Fibonacci series when *k* = 1, you should've noticed in 1) above that the ratio of consecutive numbers converge towards the famous golden ratio, φ = (√5 + 1)/2 = 1.61803... (irrespective of the starting values of the sequence). Similarly, you should've seen for *k* = 2 or *k* = 3 that a similar convergence to other irrational numbers takes place. I've taken the libery here of calling these limiting ratios the 'precious metal' ratios.

17) Click on the '**Link to graph of precious metal ratios**' button to navigate to a new sketch, which shows the graph of a polynomial function *y* = *x*^{k + 1} - *x*^{k} - 1 in terms of *k*. The positive roots of this polynomial, for *k* = 1, 2, 3, ... etc., give the corresponding values of the precious metal ratios.

18) Drag *A* along the graph to its intersection with the *x*-axis to find the approximate positive root. Compare the approximate root with the converging ratio on the 1st page when *k* = 1.

19) Click in the '**blue box**' next to *k* = 1 to change its value. Now repeat 18) above to find the approximate root when *k* = 2, and compare with the converging ratio on the 2nd page.

20) Repeat 19) above by changing the value of *k* to 3.

21) **Challenge**: Can you explain why (prove that) the roots of this polynomial, for *k* = 1, 2, 3, ... etc., give the corresponding values of the 'precious metal' ratios?

**Historical Background to Kendal's theorem**

The above generalization of the Fibonacci series started with the open-ended classroom investigation around 1988/89 of a very gifted & inspirational South African mathematics teacher, Tiekie de Jager, from Rondebosch Boy's High in Cape Town (see De Jager, 1990). However, the original proof of the generalization produced by a Grade 11 student, Shannon Kendal, had some errors. This subsequently led to further investigation and a kind of critical, 'proof-refutation' discussion in the style of Lakatos (1976) in the Letters Column of the journal *Pythagoras*, in which it had been published (see De Villiers, 2000a). Lastly, it is also possible to formulate a dual result for this particular generalization involving multiplication between terms instead of addition (see De Villiers, 2000b). This whole episode serves, once again, as an excellent example of how creative one's students can be if given the opportunity & freedom to explore, discover & prove their own observations in mathematics.

**Explore More Variations**

The Fibonacci sequence & series has an extremely vast number of interesting properties, and numerous variations & generalizations are possible.

Explore some variations on your own using a spreadsheet or suitable dynamic geometry software:

i) Try, for example, different variations of the construction rule above such as *T*_{n} + *T*_{n + 1} = *T*_{n + 3}, etc.

ii) Or instead of just adding two terms, try adding three terms at a time to produce another, etc.

iii) Or try a weighted variation of the Fibonacci rule, e.g. *p**T*_{n} + *q**T*_{n + 1} = *T*_{n + 2}, etc., where *p* and *q* are integers.

iv) Take any four consecutive Fibonacci numbers, say *T*_{n}, *T*_{n + 1}, *T*_{n + 2}, *T*_{n + 3}, and letting *a* = *T*_{n}*T*_{n + 3} and *b* = 2*T*_{n+1}*T*_{n + 2}, then the number *c* = √(*a*^{2} + *b*^{2}) is always an integer - thus the numbers *a*, *b* and *c* form a Pythagorean triple (Pagni, 2002). Does this also generally hold for Kendal's generalization? Investigate this question and also consider other possible variations (compare with De Villiers, 2002).

v) Finally, search the internet for more ideas to explore on your own.

**Some References**

Brousseau, A. (1967). A Fibonacci Generalization. *The Fibonacci Quarterly*, April, pp. 171-174.

Burton, D. M. (1991). *Burton's History of Mathematics: An Introduction* (3rd Ed.). Boston: Wm. C. Brown Publishers.

De Jager, C.J. (1990). When should we use pattern? *Pythagoras*, 23(July), pp. 11-14.

De Villiers, M. (2000a). A Fibonacci Generalization: A Lakatosian example. *Mathematics in College*, pp. 10-29.

De Villiers, M. (2000b). A Fibonacci generalization and its dual. *International Journal of Mathematical Education in Science and Technology*, 31:3, pp. 464-473.

(Provides complete proofs of Kendal's theorem about the relationship between *S _{n}* and

De Villiers, M. (2002). A further Pythagorean variation on a Fibonacci theme.

De Villiers, M. (2009). Generalizing the Golden Ratio and Fibonacci.

Falcon, S. (2004). A simple proof of an interesting Fibonacci generalization.

(Provides a complete proof of the convergence of the 'precious metal ratios' above).

Lakatos, I. (1976).

Pagni, D. (2001). Fibonacci meets Pythagoras.

Panwar, Y.K.; Singh, B. & Gupta, V. K. (2014). Generalized Fibonacci Sequences and Its Properties.

Vajda, S. (1989).

Vorobiev, N.N. (2002)

(Note that the 1st problem on p. 5 of Vorobiev's book is precisely Kendal's theorem for the standard Fibonacci series (when

**Related Links**

Golden Quadrilaterals (Generalizing the concept of a golden rectangle)

Cyclic Kepler Quadrilateral Conjectures

Golden Ratio Parallel-hexagon

Number Patterns: What numbers come next?

**Some External Links**

Fibonacci sequence

'Golden Rectangle'

Golden Ratio (Wikipedia)

Golden Ratio (Cut the Knot)

Nautilus Spirals and the Meta-Golden Ratio Chi

Myth-busting the Golden Ratio

The Parthenon and the Golden Ratio: Myth or Misinformation?

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Release: 2020Q3, semantic Version: 4.8.0, Build Number: 1070, Build Stamp: 8126303e6615/20200924134412

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Created by Michael de Villiers with *WebSketchpad*, 15 Dec 2023.