(a) Fibonacci posed the following problem: Suppose that...

(a) Fibonacci posed the following problem: Suppose that rabbits live forever and that every month cach pair produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many pairs of rabbits will we have in the $n$ th month? Show that the answer is $f_{n}$ where $\left\{f_{n}\right\}$ is the Fibonacci sequence defined in Example $3(\mathrm{c})$. (b) Let $a_{n}=f_{n+1} / f_{n}$ and show that $a_{m-1}=1+1 / a_{n-2}$. Assuming that $\left\{a_{m}\right\}$ is convergent, find its limit.

(a) Fibonacci posed the following problem:
Suppose that rabbits live forever and that every month cach pair produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many pairs of rabbits will we have in the $n$ th month?
Show that the answer is $f_{n}$ where $\left\{f_{n}\right\}$ is the Fibonacci sequence defined in Example $3(\mathrm{c})$.
(b) Let $a_{n}=f_{n+1} / f_{n}$ and show that $a_{m-1}=1+1 / a_{n-2}$. Assuming that $\left\{a_{m}\right\}$ is convergent, find its limit.

Key Concepts

Fibonacci Sequence

This is a sequence of numbers where each term is the sum of the two preceding ones, typically starting with the values 0 and 1 (or in some variations 1 and 1). It is a fundamental example of a recurrence relation in mathematics and appears in numerous applications, from nature to computer science. Its recursive definition introduces students to the idea of defining sequences in terms of their earlier terms.

Recurrence Relations

Recurrence relations are equations that recursively define a sequence: each term is a function of preceding terms. In problems such as the rabbit reproduction problem, the recurrence relation captures the way populations evolve over time, illustrating how the dynamic behavior of the sequence is determined by its initial conditions and recursive rule.

Convergence of Ratios and the Golden Ratio

Studying the ratio of consecutive terms in the Fibonacci sequence leads to the discovery of a limit, which is the golden ratio. As the sequence grows, the ratio of a term to its preceding term approaches approximately 1.618, an irrational number with many interesting properties in mathematics, art, and nature. This demonstrates not only convergence in sequences but also the deep connection between recursion and algebraic equations.

Transcript

00:01 The vibonacci sequence is defined recursively by the next conditions where the first term is one, the second term is one, and then it's built from there that the nth term is the sum of the previous two terms for every term after the third one and beyond.

00:24 So it develops the sequence one, one, one plus one is two, one plus two is three, two plus three, three plus five, three plus five, eight, five plus six.

00:32 8 is 13, 8 plus 13 is 21, and so on.

00:39 And so let's, if we suppose that rabbits live forever and that every month each pair of rabbits, each pair produces a new pair which becomes productive at age two months, if we start with one newborn pair, how many pairs of rabbits will we have in the end month? so if we were to visualize this, the first month we would have one pair, one newborn pair.

01:16 In the second month, we would have one pair which is one month old.

01:23 In the third month, we would have a pair which is reproductive and one newborn.

01:33 In the fourth month, we would have one pair which is reproductive and one month old.

01:40 I should be color coding this.

01:43 So it would make it easier.

01:47 So we've got the one pair that is got one month old and then produced a new pair.

02:00 In the fifth month, we've got the one pair that's older and had the one month old.

02:12 Now our newborn now they're one month old and we've got one pair that could have another pair.

02:35 In the sixth month i think this might actually be easier just to describe instead of trying to draw a diagram but in the first month we'll have a newborn pair in the second month we'll have a pair which is one month old in the third month, we'll have a pair which is reproductive and one newborn.

03:15 In the fourth month, we have a pair which is reproductive, a pair which is one month old, and a newborn pair.

03:23 In the fifth month, we have two pairs which is reproductive, one pair that's one month old and two newborn pairs.

03:30 In the fifth month, we have three pairs which is reproductive, two pair which is one month old, and three newborn pairs...

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