Problem 93 - College Algebra with Modeling and Visualization

Question

The Fibonacci sequence dates back to 1202 . It is one of the most famous sequences in mathematics and can be defined recursively by $$\begin{array}{l}a_{1}=1, a_{2}=1 \\ a_{n}=a_{n-1}+a_{n-2} \quad \text { for } n>2 \\ \text { (a) Find the first } 12 \text { terms of this sequence. } \\\ \text { (b) Compute } \frac{a_{n}}{a_{n-1}} \text { when } n=2,3,4, \ldots, 12 . \text { What hap- } \\ \text { pens to this ratio? }\end{array}$$ (c) Show that for \(n=2,3,\) and 4 the terms of the Fibonacci sequence satisfy the equation $$a_{n-1} \cdot a_{n+1}-a_{n}^{2}=(-1)^{n} \text { . }$$

Step-by-Step Solution

Answer

(a) First 12 terms: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. (b) Ratio approaches \(1.618\). (c) The quadratic identity holds for \(n=2,3,4\).

Step 1: Define Fibonacci sequence

The Fibonacci sequence is defined by the initial conditions: \[ a_1 = 1, \ a_2 = 1 \]and the recurrence relation: \[ a_n = a_{n-1} + a_{n-2} \text{ for } n > 2 \]

Step 2: Calculate the first 12 terms

Start with the initial terms and apply the recurrence relation repeatedly:- \( a_1 = 1 \)- \( a_2 = 1 \)- \( a_3 = a_2 + a_1 = 1 + 1 = 2 \)- \( a_4 = a_3 + a_2 = 2 + 1 = 3 \)- \( a_5 = a_4 + a_3 = 3 + 2 = 5 \)- \( a_6 = a_5 + a_4 = 5 + 3 = 8 \)- \( a_7 = a_6 + a_5 = 8 + 5 = 13 \)- \( a_8 = a_7 + a_6 = 13 + 8 = 21 \)- \( a_9 = a_8 + a_7 = 21 + 13 = 34 \)- \( a_{10} = a_9 + a_8 = 34 + 21 = 55 \)- \( a_{11} = a_{10} + a_9 = 55 + 34 = 89 \)- \( a_{12} = a_{11} + a_{10} = 89 + 55 = 144 \)The first 12 terms are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.

Step 3: Compute the ratio \( \frac{a_n}{a_{n-1}} \)

Calculate the ratio for each \( n \) from 2 to 12:- \( \frac{a_2}{a_1} = \frac{1}{1} = 1 \)- \( \frac{a_3}{a_2} = \frac{2}{1} = 2 \)- \( \frac{a_4}{a_3} = \frac{3}{2} = 1.5 \)- \( \frac{a_5}{a_4} = \frac{5}{3} \approx 1.67 \)- \( \frac{a_6}{a_5} = \frac{8}{5} = 1.6 \)- \( \frac{a_7}{a_6} = \frac{13}{8} = 1.625 \)- \( \frac{a_8}{a_7} = \frac{21}{13} \approx 1.615 \)- \( \frac{a_9}{a_8} = \frac{34}{21} \approx 1.619 \)- \( \frac{a_{10}}{a_9} = \frac{55}{34} \approx 1.618 \)- \( \frac{a_{11}}{a_{10}} = \frac{89}{55} \approx 1.6182 \)- \( \frac{a_{12}}{a_{11}} = \frac{144}{89} \approx 1.6179 \)As \( n \) increases, the ratio approaches the golden ratio \( \phi \approx 1.618 \).

Step 4: Verify the quadratic identity for \(n=2\)

For \( n=2 \): \[ a_1 \cdot a_3 - a_2^2 = 1 \cdot 2 - 1^2 = 2 - 1 = 1 \] Compare with \((-1)^2 = 1 \).The equation holds.

Step 5: Verify the quadratic identity for \(n=3\)

For \( n=3 \): \[ a_2 \cdot a_4 - a_3^2 = 1 \cdot 3 - 2^2 = 3 - 4 = -1 \] Compare with \((-1)^3 = -1 \).The equation holds.

Step 6: Verify the quadratic identity for \(n=4\)

For \( n=4 \): \[ a_3 \cdot a_5 - a_4^2 = 2 \cdot 5 - 3^2 = 10 - 9 = 1 \] Compare with \((-1)^4 = 1 \).The equation holds.

Key Concepts

Recursive Sequences

A recursive sequence is a sequence in which each term is defined based on the preceding terms. This type of sequence is fundamental in mathematics because it provides a way to build complex patterns from simple rules.
The Fibonacci sequence is a classic example of a recursive sequence. It begins with two initial terms, both equal to 1. Following these, each subsequent term is the sum of the two preceding terms. This recursive definition creates an elegant and predictable pattern.
Understanding recursive sequences is incredibly useful because they appear in various mathematical and real-world contexts. For instance, they can model population growth, financial growth, and even patterns found in nature like spirals of shells or the arrangement of leaves on a stem. Recursive sequences show how simple beginnings can lead to intricate structures.

Golden Ratio

The golden ratio, often represented by the Greek letter \( \phi \), is a special number approximately equal to 1.618. It arises from dividing one term in the Fibonacci sequence by its preceding term. As you compute the ratios of Fibonacci numbers, you will notice that they get closer and closer to \( \phi \).
This ratio is not only significant in mathematics but also in art, architecture, and nature. The Parthenon in Athens, the pyramids in Egypt, and even the proportions of the human body exhibit golden ratio principles.
Mathematically, the golden ratio is thought to be aesthetically pleasing, leading to its prevalence in design and composition. It is also significant in the field of algebra, playing a key role in the solution of quadratic equations.

Mathematical Identities

Mathematical identities are equations that are true for all values of the variables involved. They provide a reliable way to verify or derive mathematical properties. In the case of the Fibonacci sequence, there is an identity which you can verify: \[ a_{n-1} \cdot a_{n+1} - a_n^2 = (-1)^n \] This identity holds true for certain terms in the sequence and illustrates a deeper connection among the numbers.
Mathematical identities, like this one, are essential tools in algebra and number theory. They help mathematicians establish relationships between numbers and simplify complex calculations. By understanding identities, students can gain a deeper comprehension of how numbers and operations interact.