The Simplest Properties of Fibonacci Numbers

Nicolai N. Vorobiew
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Nicolai N. Vorobiew: Russian Academy of Sciences

Chapter Chapter 1 in Fibonacci Numbers, 2002, pp 5-50 from Springer

Abstract: Abstract We begin this chapter by calculating the sum of the first n Fibonacci numbers. Specifically, we are going to prove that 1.1 $$ {u_1} + {u_2} + \cdots + {u_n} = {u_{n + 2}} - 1. $$ 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

Date: 2002
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DOI: 10.1007/978-3-0348-8107-4_2

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