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. ...
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
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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 \( x^2 - x - 1 = 0 \).)*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a4cecdd-94f1-45ae-b6d2-43e39cc00a4c%2Ff6a33d14-45e7-4e36-8f68-6c0e78c89f5d%2Fm5wzb4_processed.png&w=3840&q=75)
Transcribed Image Text:### 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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