7. Use the PCI to prove the following properties of Fibonacci numbers: (d) (Binet's formula) Let o be the positive solution and p the negative solution to the equation x 2 = x + 1. (The values are o = 1+V5 and p =.) Show for all natural 1-V5 2 2 numbers n that fn $ – pn ф-р
7. Use the PCI to prove the following properties of Fibonacci numbers: (d) (Binet's formula) Let o be the positive solution and p the negative solution to the equation x 2 = x + 1. (The values are o = 1+V5 and p =.) Show for all natural 1-V5 2 2 numbers n that fn $ – pn ф-р
Advanced Engineering Mathematics
10th Edition
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![### Binet's Formula for Fibonacci Numbers
#### Problem Statement:
Use the Principle of Complete Induction (PCI) to prove the following properties of Fibonacci numbers:
#### Part (d) (Binet's Formula)
Let \( \phi \) be the positive solution and \(\rho\) the negative solution to the equation:
\[ x^2 = x + 1. \]
The values are:
\[ \phi = \frac{1 + \sqrt{5}}{2} \]
and
\[ \rho = \frac{1 - \sqrt{5}}{2}. \]
Show that for all natural numbers \(n\):
\[ f_n = \frac{\phi^n - \rho^n}{\phi - \rho}. \]
This formula provides a closed-form expression for the \(n\)-th Fibonacci number. The approach involves proving that this formula generates the Fibonacci sequence through mathematical induction.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9381f9ec-4907-481a-8789-2e44e2a64edb%2F15307d27-11ea-4d0f-a373-95d99e031cd8%2Fduba5js.png&w=3840&q=75)
Transcribed Image Text:### Binet's Formula for Fibonacci Numbers
#### Problem Statement:
Use the Principle of Complete Induction (PCI) to prove the following properties of Fibonacci numbers:
#### Part (d) (Binet's Formula)
Let \( \phi \) be the positive solution and \(\rho\) the negative solution to the equation:
\[ x^2 = x + 1. \]
The values are:
\[ \phi = \frac{1 + \sqrt{5}}{2} \]
and
\[ \rho = \frac{1 - \sqrt{5}}{2}. \]
Show that for all natural numbers \(n\):
\[ f_n = \frac{\phi^n - \rho^n}{\phi - \rho}. \]
This formula provides a closed-form expression for the \(n\)-th Fibonacci number. The approach involves proving that this formula generates the Fibonacci sequence through mathematical induction.
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