Problem 23 - A Graphical Approach to College Algebra

Question

Find the first four terms of each sequence. $$a_{1}=1, a_{2}=1, a_{n}=a_{n-1}+a_{n-2}, \text { for } n \geq 3$$ (the Fibonacci sequence)

Step-by-Step Solution

Answer

The first four terms are: 1, 1, 2, 3.

Step 1: Understand the Sequence Definition

The sequence is defined such that the first term \(a_1\) is 1 and the second term \(a_2\) is 1. For any term \(a_n\) where \(n \geq 3\), the term is the sum of the two previous terms: \(a_n = a_{n-1} + a_{n-2}\). This means we use the previous two values to calculate the new term.

Step 2: Calculate the First Two Terms

We are given that the first term \(a_1\) is 1 and the second term \(a_2\) is also 1. Therefore, the sequence starts as: \(1, 1, ...\).

Step 3: Calculate the Third Term

Since \(a_3 = a_2 + a_1\), substitute the known values \(a_2 = 1\) and \(a_1 = 1\) to find \(a_3 = 1 + 1 = 2\). Thus, the sequence so far is: \(1, 1, 2, ...\).

Step 4: Calculate the Fourth Term

To find \(a_4\), use the formula with \(a_3\) and \(a_2\): \(a_4 = a_3 + a_2 = 2 + 1 = 3\). Thus, the first four terms of the sequence are: \(1, 1, 2, 3\).

Key Concepts

Sequence Definition

Sequences are ordered lists of numbers that follow a specific pattern or rule. In mathematics, they represent a list of numbers in a specific order, where each number is called a term. Understanding sequences is crucial because they form the basis for more complex mathematical concepts.
A sequence can be defined explicitly or recursively. An explicit definition would directly calculate the term's value based on its position. A recursive definition provides a formula that relates each term to the previous terms. This recursive nature adds an element of continuity and dependence between terms.
In the Fibonacci sequence, the first two terms are defined, and each subsequent term is computed using the previous two terms. This showcases the essence of a sequence's definition as it lays out a clear pattern to follow.

Recursive Formula

A recursive formula is a mathematical way to express each term in a sequence in relation to the terms before it. It describes how to calculate a term based on its predecessors. Knowing the initial conditions and understanding the recursive formula is essential for generating a sequence.
For the Fibonacci sequence, the recursive formula is:

• \(a_1 = 1\)
• \(a_2 = 1\)
• For all \(n \geq 3\), \(a_n = a_{n-1} + a_{n-2}\)

With these formulas, you know how to start the sequence and continue it indefinitely. Each term depends on the sum of the two previous terms, capturing the recursion nature.
This formula helps not only in calculating values but also emphasizes the dependency and building process inherent in recursive sequences.

Mathematical Sequences

Mathematical sequences form a significant part of both mathematics and applied sciences, providing a way to model growth patterns, natural phenomena, and more. They are ordered arrangements of numbers governed by specific rules.
There are many types of sequences, with arithmetic and geometric sequences being common types. The Fibonacci sequence is an example of a recursive sequence, revealing an entirely different progression - each term is constructed from previous terms.
Understanding sequences involve recognizing patterns, employing formulas, and sometimes deriving rules from observed regularities. They provide a structured approach to anticipate future terms and analyze trends.
Thus, sequences are everywhere around us; knowing their principles assists in unlocking many mathematical concepts.