Problem 52 - Calculus Early Transcendentals: Pearson New International Edition

Question

A famous sequence \(\left\\{f_{n}\right\\}\), called the Fibonacci Sequence after Leonardo Fibonacci, who introduced it around A.D. 1200 , is defined by the recursion formula $$ f_{1}=f_{2}=1, \quad f_{n+2}=f_{n+1}+f_{n} $$ (a) Find \(f_{3}\) through \(f_{10}\) - (b) Let \(\phi=\frac{1}{2}(1+\sqrt{5}) \approx 1.618034\). The Greeks called this number the golden ratio, claiming that a rectangle whose dimensions were in this ratio was "perfect." It can be shown that $$ \begin{aligned} f_{n} &=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right] \\\ &=\frac{1}{\sqrt{5}}\left[\phi^{n}-(-1)^{n} \phi^{-n}\right] \end{aligned} $$ Check that this gives the right result for \(n=1\) and \(n=2\). The general result can be proved by induction (it is a nice challenge). More in line with this section, use this explicit formula to prove that \(\lim _{n \rightarrow \infty} f_{n+1} / f_{n}=\phi\). (c) Using the limit just proved, show that \(\phi\) satisfies the equation \(x^{2}-x-1=0\). Then, in another interesting twist, use the Quadratic Formula to show that the two roots of this equation are \(\phi\) and \(-1 / \phi\), two numbers that occur in the explicit formula for \(f_{n}\).

Step-by-Step Solution

Answer

(a) Fibonacci values are 2, 3, 5, 8, 13, 21, 34, 55. (b) The explicit formula correctly gives 1 for \(n=1\) and \(n=2\). Limit \(f_{n+1}/f_{n} = \phi\) as \(n\to\infty\). (c) \(\phi\) satisfies \(x^2-x-1=0\); roots are \(\phi\) and \(-1/\phi\).

Step 1: Calculate Fibonacci Values

The Fibonacci sequence is defined by the formula \(f_{n+2} = f_{n+1} + f_{n}\) with initial conditions \(f_{1} = 1\) and \(f_{2} = 1\). Thus:- \(f_{3} = f_{2} + f_{1} = 1 + 1 = 2\)- \(f_{4} = f_{3} + f_{2} = 2 + 1 = 3\)- \(f_{5} = f_{4} + f_{3} = 3 + 2 = 5\)- \(f_{6} = f_{5} + f_{4} = 5 + 3 = 8\)- \(f_{7} = f_{6} + f_{5} = 8 + 5 = 13\)- \(f_{8} = f_{7} + f_{6} = 13 + 8 = 21\)- \(f_{9} = f_{8} + f_{7} = 21 + 13 = 34\)- \(f_{10} = f_{9} + f_{8} = 34 + 21 = 55\).

Step 2: Verify Explicit Formula for Small Cases

To verify that \(f_{n} = \frac{1}{\sqrt{5}}\left[(\frac{1+\sqrt{5}}{2})^{n} - (\frac{1-\sqrt{5}}{2})^{n}\right]\) gives correct results for \(n=1\) and \(n=2\),For \(n=1\):- \(f_{1} = \frac{1}{\sqrt{5}}\left[\phi - (-1)\cdot\phi^{-1}\right] = 1\).For \(n=2\):- \(f_{2} = \frac{1}{\sqrt{5}}\left[\phi^2 - \phi^{-2}\right] = 1\).

Step 3: Prove Limit of Fibonacci Ratio

To prove \(\lim_{n \to \infty}\frac{f_{n+1}}{f_{n}} = \phi\), notice that as \(n\) becomes very large, \((-1/\phi)^{n}\) approaches zero because \(\left| -1/\phi \right| < 1\). Thus, from the formula:\[\frac{f_{n+1}}{f_{n}} = \frac{\phi^{n+1} - (-1/\phi)^{n+1}}{\phi^{n} - (-1/\phi)^{n}} \approx \phi\]As \((-1/\phi)^{n+1}, (-1/\phi)^{n} \to 0\).

Step 4: Verify Golden Ratio Quadratic Equation

Given \(\phi = \frac{1+\sqrt{5}}{2}\), substitute into the equation \(x^{2} - x - 1 = 0\):\((\phi)^{2} - \phi - 1 = 0\)\[\left(\frac{1+\sqrt{5}}{2}\right)^{2} - \frac{1+\sqrt{5}}{2} - 1 = 0\]Simplifying verifies the equation holds true, confirming \(\phi\).

Step 5: Find Roots Using Quadratic Formula

The quadratic formula \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) applied to \(x^{2} - x - 1 = 0\) gives:\(\phi = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2}\)Thus, the roots are \(\phi = \frac{1+\sqrt{5}}{2}\) and \(-1/\phi\). These roots match those in the explicit formula for \(f_{n}\).

Key Concepts

Golden Ratio

The Golden Ratio, denoted by the symbol \( \phi \), is approximately 1.618034. This number has fascinated mathematicians, artists, and architects for centuries due to its unique properties and aesthetic appeal. The Greeks considered a rectangle with this ratio to possess perfect proportions.
This ratio frequently appears in nature, art, and architecture, contributing to its mystique and beauty.To understand how the Golden Ratio relates to the Fibonacci Sequence, note the formula:\[\phi = \frac{1+\sqrt{5}}{2}\]The Golden Ratio also appears as the limit of the ratio of successive Fibonacci numbers, specifically:\[\lim _{n \rightarrow \infty} \frac{f_{n+1}}{f_{n}} = \phi\] This remarkable property shows the deep connection between the Fibonacci Sequence and the Golden Ratio. As the sequence progresses, the ratio of consecutive Fibonacci numbers approaches \( \phi \), illustrating its pervasive presence in patterns and growth sequences.

Recursion Formula

Recursion formulas are powerful tools in mathematics, allowing us to define sequences or patterns using previous terms. In the Fibonacci Sequence, the recursion formula is:\[f_{n+2} = f_{n+1} + f_{n}\]This means each term is the sum of the two preceding ones.
Starting values, called initial conditions, determine the sequence's specific path. For the Fibonacci Series:

• \(f_1 = 1\)
• \(f_2 = 1\)

These values generate a unique and widely known sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on. This sequence is found in diverse fields, from computer algorithms to biological settings.
Recursion is vital for calculating complex sequences efficiently, as it eliminates the need to compute every term individually, leveraging earlier computed values instead.

Quadratic Formula

The quadratic formula is a handy method for solving quadratic equations, such as \(x^2 - x - 1 = 0\). The quadratic formula provides solutions for equations in the form \(ax^2 + bx + c = 0\), and is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]For the quadratic equation related to the Golden Ratio:

• \(a = 1\)
• \(b = -1\)
• \(c = -1\)

Using these coefficients, the solutions are:\[x = \frac{1 \pm \sqrt{5}}{2}\]These solutions reveal that \(\phi = \frac{1+\sqrt{5}}{2}\) and its counterpart \(-\frac{1}{\phi}\) are the roots of the equation. These values not only correspond with the Golden Ratio but also feature in the explicit formula for calculating Fibonacci numbers.

Limit of a Sequence

In mathematics, the limit of a sequence is a value that the elements of the sequence approach as the index (usually \(n\)) goes to infinity. It helps us understand the long-term behavior of sequences.
For the Fibonacci Ratio, consider the sequence of ratios:\[\frac{f_{n+1}}{f_{n}}\]As \(n\) becomes very large,

• \(\left( -\frac{1}{\phi} \right)^{n}\) approaches zero
• leaving the ratio itself approaching the Golden Ratio \(\phi\).

This limit shows a fascinating connection between the recursive pattern and \(\phi\), illustrating the predictive power of limits in sequences. Understanding limits helps determine steady states, predict future behavior, and model natural phenomena accurately.