Purchase Solution

Fibonacci Sequences, Convergence and Limits

Not what you're looking for?

(a) find the first 12 terms of the Fibonacci sequence Fn defined by the Fibonacci relationship

Fn=Fn-1+Fn-2

where F1=1, F2=1.

(b) Show that the ratio of successive F's appears to converge to a number satisfying r2=r+1.

(c) Let r satisfy r2=r+1. Show that the sequence sn=Arn, where A is any constant, satisfies the Fibonacci relationship. Is the Fibonnaci sequence that you found in (a) given by this formula for some A?

(d) Notice that the quadratic equation satisfied by r above has two roots. Let them be r1 and r2, and show that Ar1n+Br2n satisfies the Fibonnaci relationship, for any choice of constants A and B.

(e) Use the observation in (d) to find a formula for the nth term of the Fibonnaci sequence, and prove that it works.

Solution Summary

Fibonacci Sequences, Convergence and Limits are investigated. The solution is detailed and well presented.

Solution Preview

Hello and thank you for posting your question to Brainmass!

The solution is attached below (next to the paperclip icon) in two formats. one is in Word XP Format, while the other is in Adobe pdf format. Therefore you can choose ...

Free BrainMass Quizzes

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.

This quiz test you on how well you are familiar with solving quadratic inequalities.

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.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts