A Fibonacci Generalization - Kendal's theorem

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 Da Vinci Code (2003) by Dan Brown. Though mathematically fascinating, it is unfortunately a common myth that the Fibonacci series (and the golden ratio) somehow has mystical, magical or divine properties. It is just one particular mathematical series among an infinitude of other possible series - as the investigation below will hopefully show.

Explore
In the interactive sketch below the first twenty terms Tn of the Fibonacci sequence were produced with the rule Tn + Tn + 1 = Tn + 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, Sn, in the 5th column with the terms Tn of the Fibonacci sequence in the 3rd column. What do you notice? Can you see a pattern and formulate it?
3) Click in the 'blue box' next to the seed T1 = 0 to change the starting value. Now carefully check again your observations in 2) and 3) above.
4) If necessary, repeat step 3), then formulate your observations in 1) and 2) above.
(Click on the 'Show Pattern' and 'Show Formulate Pattern' buttons if you need help or just want to check your conclusion in regard to 2) above.)

.sketch_canvas { border: medium solid lightgray; display: inline-block; } 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 Tn + Tn + 2 = Tn + 3.
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 n terms, Sn, in the 6th column with the terms Tn of the sequence in the 3rd column. What do you notice? Can you see a pattern and formulate it?
7) Click in the 'blue box' next to the seed T1 = 0 to change the starting value. Now carefully check again your observations in 5) and 6) above.
8) If necessary, repeat step 7), then formulate your observations in 5) and 6) above.
(Click on the 'Show Pattern' and 'Show Formulate Pattern' buttons if you need help or just want to check your conclusion in regard to 6) above.)

b) Click on the 'Link to k = 3' button to navigate to a new sketch, which shows a new sequence & series created with the rule Tn + Tn + 3 = Tn + 4.
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 n terms, Sn, in the 7th column with the terms Tn of the sequence in the 3rd column. What do you notice? Can you see a pattern and formulate it?
11) Note that for this page it is unfortunately not possible to edit the starting value or seed T1.
12) Formulate your observations in 9) and 10) above.
(Click on the 'Show Pattern' and 'Show Formulate Pattern' buttons if you need help or just want to check your conclusion in regard to 10) above.)

Generalization
13) Can you now generalize the Fibonacci rule Tn + Tn + 1 = Tn + 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 Sn and Tn + k + 1.
15) What about a converse of your conjecture in 14) regarding the relationship between Sn and Tn + k + 1? Explore by hand calculation or use suitable software!

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 = xk + 1 - xk - 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 Tn + Tn + 1 = Tn + 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. pTn + qTn + 1 = Tn + 2, etc., where p and q are integers.
iv) Take any four consecutive Fibonacci numbers, say Tn, Tn + 1, Tn + 2, Tn + 3, and letting a = TnTn + 3 and b = 2Tn+1Tn + 2, then the number c = √(a2 + b2) 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 Sn and Tn + k + 1).
De Villiers, M. (2002). A further Pythagorean variation on a Fibonacci theme. Mathematics in School, 31(5), p. 22.
De Villiers, M. (2009). Generalizing the Golden Ratio and Fibonacci. Learning and Teaching Mathematics, No. 7, pp. 39-41.
Falcon, S. (2004). A simple proof of an interesting Fibonacci generalization. International Journal of Mathematical Education in Science and Technology, 35:2, pp. 259-261.
(Provides a complete proof of the convergence of the 'precious metal ratios' above).
Lakatos, I. (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge: Cambridge University Press.
Pagni, D. (2001). Fibonacci meets Pythagoras. Mathematics in School, 30(4), pp. 39-40.
Panwar, Y.K.; Singh, B. & Gupta, V. K. (2014). Generalized Fibonacci Sequences and Its Properties. Palestine Journal of Mathematics, Vol. 3(1), pp. 141–147.
Vajda, S. (1989). Fibonacci & Lucas Numbers, and the Golden Section. New York: Halsted Press.
Vorobiev, N.N. (2002) Fibonacci Numbers. Basel: Birkhäuser Verlag.
(Note that the 1st problem on p. 5 of Vorobiev's book is precisely Kendal's theorem for the standard Fibonacci series (when k = 1)).

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