Fibonacci Numbers and Golden Rule
Not what you're looking for?
Show that |(frac{f_(n+1)}{f_n}) - phi| = frac{1}{(f_n)(phi^{n+1})}
and lim_{n --> infty} frac{f_{n+1}}{f_n} = phi,
where phi is the Golden Ration and is the unique positive root of phi^2 - phi - 1 = 0.
For some discussion on this question, see http://math.stackexchange.com/questions/106049/another-way-to-go-about-proving-the-limit-of-fibonaccis-sequence-quotient?lq=1
For a power series involving the nth fibonacci number f_n: Define f(z) = sum_{n = 0}^infty (f_n)(z^n). Show the the power series has a radius of convergence R = frac{1}{phi} and for |z| < R, f(z) = frac{1}{1 - z - z^2}
Purchase this Solution
Solution Summary
The expert examines Fibonacci Numbers and the Golden Rule.
Solution Preview
See attachment for full solutions.
The recurrence formula of Fibonacci sequences can be written as:
f_n+1 - f_n - f_n-1 = 0
Searching for solutions of form
f_n = C*r^n ...
Purchase this Solution
Free BrainMass Quizzes
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.