Question

Give the first five terms of the given sequence. $$\left\\{d_{n}\right\\}=\left\\{\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right)\right\\}$$

Short answer

The first five terms are 1, 1, 2, 3, 5.

Step-by-step solution

Understanding the Sequence Formula

The given sequence \({d_n}\) is defined as \(d_n = \frac{1}{\sqrt{5}} \left( \left(\frac{1+\sqrt{5}}{2}\right)^{n} - \left(\frac{1-\sqrt{5}}{2}\right)^{n} \right)\). This formula is similar to Binet's formula for the Fibonacci sequence, where \(d_n\) should resemble the Fibonacci number \(F_n\).

Calculate the First Term

Substitute \(n=1\) in the formula: \[d_1 = \frac{1}{\sqrt{5}} \left( \left(\frac{1+\sqrt{5}}{2}\right)^{1} - \left(\frac{1-\sqrt{5}}{2}\right)^{1} \right) = 1.\] Therefore, the first term \(d_1 = 1\).

Calculate the Second Term

Substitute \(n=2\) in the formula: \[d_2 = \frac{1}{\sqrt{5}} \left( \left(\frac{1+\sqrt{5}}{2}\right)^{2} - \left(\frac{1-\sqrt{5}}{2}\right)^{2} \right) = 1.\] So, the second term \(d_2 = 1\).

Calculate the Third Term

Substitute \(n=3\) in the formula: \[d_3 = \frac{1}{\sqrt{5}} \left( \left(\frac{1+\sqrt{5}}{2}\right)^{3} - \left(\frac{1-\sqrt{5}}{2}\right)^{3} \right) = 2.\] Thus, the third term \(d_3 = 2\).

Calculate the Fourth Term

Substitute \(n=4\) in the formula: \[d_4 = \frac{1}{\sqrt{5}} \left( \left(\frac{1+\sqrt{5}}{2}\right)^{4} - \left(\frac{1-\sqrt{5}}{2}\right)^{4} \right) = 3.\] Therefore, the fourth term \(d_4 = 3\).

Calculate the Fifth Term

Substitute \(n=5\) in the formula: \[d_5 = \frac{1}{\sqrt{5}} \left( \left(\frac{1+\sqrt{5}}{2}\right)^{5} - \left(\frac{1-\sqrt{5}}{2}\right)^{5} \right) = 5.\] Thus, the fifth term \(d_5 = 5\).

Compile the First Five Terms

The first five terms of the sequence \({d_n}\) are 1, 1, 2, 3, and 5.

Key concepts

Binet's Formula

Understanding Binet's formula is crucial to comprehend the Fibonacci sequence deeply. Binet's formula for the Fibonacci sequence provides a way to express the Fibonacci numbers with a closed formula. The formula is: \[F_n = \frac{1}{\sqrt{5}} \left( \left(\frac{1 + \sqrt{5}}{2}\right)^n - \left(\frac{1 - \sqrt{5}}{2}\right)^n \right)\]This might seem complex initially, but it represents an elegant mathematical trick to compute Fibonacci numbers without iterative methods.

• The term \(\frac{1 + \sqrt{5}}{2}\) is known as the golden ratio \(\phi\).
• The term \(\frac{1 - \sqrt{5}}{2}\) is its conjugate, often labeled as \(\psi\).

The role of Binet's formula is to derive the nth Fibonacci number directly by plugging in the value of \(n\). Unlike its recursive definition, Binet's equation shows how Fibonacci numbers can substantially grow as a function of \(n\). The negative term diminishes effectively to zero in practical applications due to its exponential decay. Thus, Binet's formula simplifies directly to exhibiting the Fibonacci number growth governed mainly by the powers of the golden ratio.

Sequence Formula

A sequence formula defines each term of a sequence, allowing us to calculate any term if its position is known. In the exercise's sequence, the formula is: \[d_n = \frac{1}{\sqrt{5}} \left( \left(\frac{1 + \sqrt{5}}{2}\right)^n - \left(\frac{1 - \sqrt{5}}{2}\right)^n \right)\]This formula works similarly to Binet's formula due to its resemblance.

• The sequence in question is representative of Fibonacci numbers.
• You input the position \(n\) to find the value \(d_n\).

Calculating terms directly using this formula involves substituting values of \(n\). The outputs show how each term, such as 1, 1, 2, 3, and 5 in this exercise, form the initial Fibonacci sequence. The power terms \((n)\) are calculated to uncover the sequence, demonstrating an exponential nature similar to how Fibonacci numbers progress.

Integer Sequence

Integer sequences are a vital part of mathematics, offering insight into how numbers can be systematically organized and applied practically. They consist of numbers arranged in a specific, deterministic order based on a given rule or formula.

• Each number in the sequence is called a "term."
• An integer sequence can be finite or infinite.

This exercise uses a sequence starting with integers 1, 1, 2, 3, and 5. It's not only a list of numbers but a predictable pattern—the essence of the Fibonacci sequence. Integer sequences are perfect for modeling growth in natural systems, making them useful in both mathematical theory and real-world applications. Understanding how these sequences work can unlock deeper insights into the mathematical structures underlying many patterns in the world.