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From golden-ratio equalities to Fibonacci and Lucas identities

Published online by Cambridge University Press:  23 January 2015

Martin Griffiths
Martin Griffiths*
Affiliation:
Mathematical Institute, University of Oxford OX1 3LB
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Extract

We demonstrate here a remarkably simple method for deriving a large number of identities involving the Fibonacci numbers, Lucas numbers and binomial coefficients. As will be shown, this is based on the utilisation of some straightforward properties of the golden ratio in conjunction with a result concerning irrational numbers. Indeed, for the simpler cases at least, the derivations could be understood by able high-school students. In particular, we avoid the use of exponential generating functions, matrix methods, Binet's formula, involved combinatorial arguments or lengthy algebraic manipulations.

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Type
Articles
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The Mathematical Gazette , Volume 97 , Issue 539 , July 2013 , pp. 234 - 241
Copyright
Copyright © The Mathematical Association 2013

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