Problem 68 - Precalculus: Mathematical for Calculus

Question

Define the sequence $$G_{n}=\frac{1}{\sqrt{5}}\left(\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}}\right)$$

Step-by-Step Solution

Answer

\(G_n\) is the \(n\)-th Fibonacci number.

Step 1: Understanding the Sequence Definition

The sequence \(G_n\) is defined as: \[G_{n} = \frac{1}{\sqrt{5}}\left(\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}}\right)\]This formula is a closed-form expression which is derived from linear recurrence relations.

Step 2: Recognizing the Formula Type

The expression for \(G_n\) resembles the closed form for the Fibonacci sequence, known as Binet’s formula. It suggests that the sequence \(G_n\) is actually equivalent to terms in the Fibonacci sequence.

Step 3: Simplify Binet’s Formula for Fibonacci Sequence

Recall that Binet’s formula for the Fibonacci sequence \(F_n\) is expressed as: \[F_{n} = \frac{1}{\sqrt{5}}\left(\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}}\right)\]This formula is identical to the definition of \(G_n\). Thus, \(G_n = F_n\).

Step 4: Draw Conclusions from the Comparison

Given that \(G_n\) takes the same form as Binet’s formula for \(F_n\), we conclude that \(G_n\) is equivalent to \(F_n\), where \(F_n\) represents the \(n\)-th Fibonacci number.

Key Concepts

Closed-form Expression

A closed-form expression is a mathematical formula that allows us to calculate terms in a sequence directly, without needing to refer to previous terms. It provides a neat and elegant way of computing values. Instead of laboriously calculating each term one-by-one, a closed-form expression uses a concise formula to find any term in the sequence.
A classic example is Binet's formula, which offers a closed-form expression for Fibonacci numbers. This formula assembles mathematical expressions into a single equation that can yield the Fibonacci number at position "n" with one calculation. This efficiency not only saves time but also simplifies understanding complex series.
In the sequence given, the expression \[G_{n} = \frac{1}{\sqrt{5}}\left(\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}}\right)\] is indeed a closed-form, because it gives the value of \(G_n\) directly from \(n\) without any iterative steps.

Sequence Definition

A sequence definition lays down the rules or an equation by which each term in the sequence can be determined. It defines how the sequence evolves and what its characteristics will be.
There are different ways to define a sequence:

• Explicit Definition: This defines each term by a direct formula. For example, the sequence for the Fibonacci numbers has an explicit definition through Binet's formula.
• Recursive Definition: This defines each term based on its predecessors. For example, the traditional Fibonacci sequence is defined recursively as \(F_n = F_{n-1} + F_{n-2}\) with initial conditions \(F_0 = 0\) and \(F_1 = 1\).

For the sequence \(G_n\), it is explicitly defined, using a formula derived from Binet’s formula. This means every term is calculated independently based on its position in the sequence rather than previous terms. This explicit definition distinguishes it from recursive sequences which need all previous values to calculate subsequent terms.

Fibonacci Sequence

The Fibonacci sequence is a famous sequence of numbers where each number is the sum of the two preceding ones. This sequence often starts with 0 and 1. Represented as \(F_0 = 0, F_1 = 1, F_2 = 1, F_3 = 2,...\), it reflects patterns that appear in natural phenomena, art, and nature.
Key points about the Fibonacci sequence include:

• Each term is generated by summing the two terms before it, \(F_n = F_{n-1} + F_{n-2}\).
• It demonstrates growth patterns seen in biological settings, like the arrangement of leaves on a stem or the spirals of a shell.
• With Binet's formula, there's an explicit calculation for each Fibonacci number, making it possible to directly find any term without referencing predecessors.

With the closed-form expression for Fibonacci numbers, such as Binet’s formula, calculations become straightforward and computationally efficient as seen in the sequence \(G_n\), which takes the same form as the Fibonacci sequence derivation.