{Use of Tech} Fibonacci sequence
The famous Fibonacci sequence was proposed by Leonardo Pisano, also known as Fibonacci, in about A.D. 1200 as a model for the growth of rabbit populations.

It is given by the recurrence relation: fₙ₊₁ = fₙ + fₙ₋₁,for n = 1, 2, 3, … where f₀ = 1 and f₁ = 1. Each term of the sequence is the sum of its two predecessors.

b.Is the sequence bounded?

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Understand the definition of the Fibonacci sequence given by the recurrence relation: \(f_{n+1} = f_n + f_{n-1}\) with initial conditions \(f_0 = 1\) and \(f_1 = 1\).

Recall that a sequence is bounded if there exists some real number \(M\) such that for all \(n\), \(|f_n| \leq M\).

Analyze the behavior of the Fibonacci sequence by considering the growth of its terms. Since each term is the sum of the two previous terms, the sequence tends to increase as \(n\) increases.

Recognize that the Fibonacci sequence grows approximately exponentially, as it can be expressed using Binet's formula involving powers of the golden ratio \(\phi = \frac{1 + \sqrt{5}}{2}\), which is greater than 1.

Conclude that because the terms grow without bound (they increase indefinitely), the Fibonacci sequence is not bounded.

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주요 개념

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Recurrence Relations

A recurrence relation defines each term of a sequence using previous terms. In the Fibonacci sequence, each term is the sum of the two preceding terms, starting with initial values. Understanding this helps analyze how the sequence evolves over time.

Boundedness of Sequences

A sequence is bounded if all its terms lie within some fixed interval. To determine if a sequence is bounded, one must check whether its terms stay below or above certain limits as the sequence progresses indefinitely.

Growth Behavior of the Fibonacci Sequence

The Fibonacci sequence grows exponentially because each term is roughly 1.618 times the previous term (the golden ratio). This means the terms increase without bound, indicating the sequence is unbounded.