### Fibonacci Sequence Definition and Problems#### Definition:The Fibonacci sequence $ F_n $ is defined as follows:- $ F_0 = 0 $- $ F_1 = 1 $- For $ n \geq 2 $: $ F_n = F_{n-1} + F_{n-2} $#### Problems to Prove:(a) **Prove for $ n \geq 0 $:**\[ F_0 + F_1 + \cdots + F_n = F_{n+2} - 1. \](b) **Prove for $ n \geq 0 $:**\[ F_1 + F_3 + \cdots + F_{2n+1} = F_{2n+2}. \](c) **Prove for $ n \geq 1 $:**\[ F_{n+1}F_{n-1} - F_n^2 = (-1)^n. \](d) **Prove using the roots of the equation:**\[ F_n = \frac{1}{\sqrt{5}} \left( p^n - q^n \right), \]where \[ p = \frac{1+\sqrt{5}}{2} \]and \[ q = \frac{1-\sqrt{5}}{2}. \]*(Hint: $ p $ and $ q $ are the two roots of the equation $ x^2 - x - 1 = 0 $.)*

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Description: ### Fibonacci Sequence Definition and Problems#### Definition:The Fibonacci sequence $ F_n $ is defined as follows:- $ F_0 = 0 $- $ F_1 = 1 $- For $ n \geq 2 $: $ F_n = F_{n-1} + F_{n-2} $#### Problems to Prove:(a) **Prove for \( n \geq 0

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The Fibonacci sequence F, is defined as follows: Fo = 0 F1 = 1 Vn 2 2: Fn = Fn-1+ Fn-2 (a) Prove that Vn 2 0: Fo + F1 + · +F = Fn+2 - 1. ...

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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### Fibonacci Sequence Definition and Problems

#### Definition:
The Fibonacci sequence $ F_n $ is defined as follows:
- $ F_0 = 0 $
- $ F_1 = 1 $
- For $ n \geq 2 $: $ F_n = F_{n-1} + F_{n-2} $

#### Problems to Prove:

(a) **Prove for $ n \geq 0 $:**
\[ F_0 + F_1 + \cdots + F_n = F_{n+2} - 1. \]

(b) **Prove for $ n \geq 0 $:**
\[ F_1 + F_3 + \cdots + F_{2n+1} = F_{2n+2}. \]

(c) **Prove for $ n \geq 1 $:**
\[ F_{n+1}F_{n-1} - F_n^2 = (-1)^n. \]

(d) **Prove using the roots of the equation:**
\[ F_n = \frac{1}{\sqrt{5}} \left( p^n - q^n \right), \]
where
\[ p = \frac{1+\sqrt{5}}{2} \]
and
\[ q = \frac{1-\sqrt{5}}{2}. \]

*(Hint: $ p $ and $ q $ are the two roots of the equation $ x^2 - x - 1 = 0 $.)*

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