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The Simplest Properties of Fibonacci Numbers

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Fibonacci Numbers
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Abstract

We begin this chapter by calculating the sum of the first n Fibonacci numbers. Specifically, we are going to prove that

$$ {u_1} + {u_2} + \cdots + {u_n} = {u_{n + 2}} - 1. $$
(1.1)

Indeed, we have

$$ \begin{array}{*{20}{c}} {{u_1} = {u_3} - {u_2},} \\ {{u_2} = {u_4} - {u_3},} \\ {{u_3} = {u_5} - {u_4},} \\ \cdots \\ {{u_{n - 1}} = {u_{n + 1}} - {u_n},} \\ {{u_n} = {u_{n + 2}} - {u_{n + 1}}.} \end{array} $$

Adding up all these equations term by term we get

$$ {u_1} + {u_2} + \cdots + {u_n} = {u_{n + 2}} - {u_2}. $$

It remains to recall that u 2 = 1

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© 2002 Springer Basel AG

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Vorobiew, N.N. (2002). The Simplest Properties of Fibonacci Numbers. In: Fibonacci Numbers. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8107-4_2

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  • DOI: https://doi.org/10.1007/978-3-0348-8107-4_2

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-7643-6135-8

  • Online ISBN: 978-3-0348-8107-4

  • eBook Packages: Springer Book Archive

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