The Fibonacci numbers F0, F1, F2, …, are defined by the rule...

The Fibonacci numbers $F_{0}, F_{1}, F_{2}, \ldots,$ are defined by the rule \[ F_{0}=0, F_{1}=1, F_{n}=F_{n-1}+F_{n-2} \] In this problem we will confirm that this sequence grows exponentially fast and obtain some bounds on its growth. (a) Use induction to prove that $F_{n} \geq 2^{0.5 n}$ for $n \geq 6$ (b) Find a constant $c<1$ such that $F_{n} \leq 2^{c n}$ for all $n \geq 0 .$ Show that your answer is correct. (c) What is the largest $c$ you can find for which $F_{n}=\Omega\left(2^{\circ n}\right) ?$

The Fibonacci numbers $F_{0}, F_{1}, F_{2}, \ldots,$ are defined by the rule
\[
F_{0}=0, F_{1}=1, F_{n}=F_{n-1}+F_{n-2}
\]
In this problem we will confirm that this sequence grows exponentially fast and obtain some bounds on its growth.
(a) Use induction to prove that $F_{n} \geq 2^{0.5 n}$ for $n \geq 6$
(b) Find a constant $c<1$ such that $F_{n} \leq 2^{c n}$ for all $n \geq 0 .$ Show that your answer is correct.
(c) What is the largest $c$ you can find for which $F_{n}=\Omega\left(2^{\circ n}\right) ?$

Key Concepts

Mathematical Induction

This is a proof technique used to establish that a statement holds for all natural numbers. In the context of recursively defined sequences like the Fibonacci numbers, induction is used to prove inequalities or properties by verifying the base case and then showing that if the statement holds for one or more consecutive values, it will hold for the next value in the sequence.

Recurrence Relations

A recurrence relation defines a sequence in terms of previous terms. The Fibonacci sequence is a classic example with the rule F? = F??? + F???. Understanding recurrence relations is crucial for analyzing the behavior, growth, and properties of sequences, and they often lead to finding closed-form expressions or bounds on sequence elements.

Exponential Growth

Exponential growth describes a process where the rate of increase is proportional to the current value, leading to a rapid increase over time. In the context of Fibonacci numbers, proving that they grow at least as fast as an exponential function (e.g., 2^(0.5n)) or are bounded above by another exponential function involves comparing the recursive definition with exponential trends.

Asymptotic Notation

Asymptotic notation, such as Big Omega (?) and Big O, is used to describe the limiting behavior of functions. It provides a way to formally compare the growth rates of sequences or functions. In this problem, asymptotic notation is used to express the lower bounds (?) and upper bounds (O) for Fibonacci numbers in terms of exponential functions.

Transcript

00:01 The problem here asks you to prove exponential lower and upper bounds on the growth of fibonacci numbers.

00:08 Specifically, the question is actually to show that from some point and greater than equal to 6, a fibonacci number grows at least as fast as 2 to the 0 .5m power.

00:25 The other thing is that you find a constancy, which is less than 1, to establish an exponential upper bound.

00:32 Of where you have the tributtaccharine number is less than equal to 2 raised to the constant times n.

00:40 And the third thing is to determine the largest constant c such that that nature number fn grows at least on the order of two to the c times n.

00:54 Alright so let's go through it here.

00:58 So basically you have type of natural numbers defined by recurrent.

01:03 So you have those.

01:06 And yes, everything is type out here, so we can pause the video, rewind, and really get prepared to confirm that this sequence versus exponentially fast and it's out you we're going to establish some bounds and its growth.

01:21 So first thing is to use induction.

01:23 This is how you go with induction.

01:25 You will try with base cases first, and then you do some inductive step, right? this is how you give an inductive step.

01:35 Okay, this is a statement of the inductive step.

01:38 And then, from here, what you do is to, well, this is set defined into algebra here.

01:45 And so we see that by induction, the inequality holds for all and per year.

01:53 Okay.

01:56 That's fine.

01:57 Next thing here is to, the second part, the question is to find that constant.

02:02 So this is how we do it.

02:05 And we find this constant and to show the answer is correct...

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