Problem 25 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Known Terms

We are given that the first two terms of the sequence are \(a_{1}=1\) and \(a_{2}=1\).

Step 2: Apply the Recurrence Relation for Third Term

Using the recurrence relation \(a_{n}=a_{n-1}+a_{n-2}\), calculate the third term: \(a_{3}=a_{2}+a_{1}=1+1=2\).

Step 3: Apply the Recurrence Relation for Fourth Term

Use the recurrence relation again for the fourth term: \(a_{4}=a_{3}+a_{2}=2+1=3\).

Step 4: Compile the First Four Terms

List all the terms calculated: The first four terms are \(a_{1}=1\), \(a_{2}=1\), \(a_{3}=2\), and \(a_{4}=3\).

Key Concepts

Recurrence Relation

The concept of a recurrence relation is quite interesting and is a foundational tool in sequences like the Fibonacci sequence. It's basically a way to describe a sequence where each term is defined using one or more of its preceding terms.
In the Fibonacci sequence, the recurrence relation is defined as:

• \( a_n = a_{n-1} + a_{n-2} \) for \( n \geq 3 \).
• This means that to find any term in the sequence, you simply add the two previous terms.
• The sequence starts with initial conditions: \( a_1 = 1 \) and \( a_2 = 1 \).

This way of defining a sequence is powerful and helps us understand complex sequences by breaking them down into simpler, more manageable parts.

Sequence Terms

In mathematics, understanding terms in a sequence is crucial. Each term is a component in the series of numbers generated.
For the Fibonacci sequence, we can understand its terms as follows:

• The initial terms are always given: \( a_1 = 1 \) and \( a_2 = 1 \).
• For \( a_3 \), by using the formula \( a_3 = a_2 + a_1 \), we get \( a_3 = 1 + 1 = 2 \).
• Likewise, for \( a_4 \), applying \( a_4 = a_3 + a_2 \), results in \( a_4 = 2 + 1 = 3 \).

Each subsequent term is derived using the previous terms as per the recurrence relation, providing a systematic way to expand and understand the sequence.

Mathematical Induction

Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers. It can be especially useful when working with sequences defined by a recurrence relation. Here's a simple breakdown of how it generally works:

• **Base Case**: First, you verify the statement is true for the initial term or terms - often for \( n = 1 \) or \( n = 2 \) when dealing with sequences.
• **Inductive Step**: Next, assume the statement holds for some arbitrary term \( n = k \). Then prove it holds for \( n = k+1 \).

In the context of our sequence, we could use induction to prove that each term in the Fibonacci sequence follows the pattern of the recurrence relation continually, further solidifying our understanding of how each term is related to its predecessors.