# The Simplest Properties of 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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### Cite this chapter

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

• 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