Problem 27 - College Algebra with Modeling and Visualization

Question

Complete the following for the recursively defined sequence. (a) Find the first four terms. (b) Graph these terms. \(a_{n}=a_{n-1} a_{n-2} ; a_{1}=2, a_{2}=3\)

Step-by-Step Solution

Answer

The first four terms are 2, 3, 6, and 18.

Step 1: Identify First Term

We are given that the first term, \(a_1\), is 2. This is directly provided in the sequence definition.

Step 2: Identify Second Term

The second term, \(a_2\), is given as 3 from the problem statement. No calculations are needed here.

Step 3: Calculate Third Term

Using the recursive formula \(a_n = a_{n-1} a_{n-2}\), we find \(a_3 = a_2 \, a_1 = 3 \times 2 = 6\).

Step 4: Calculate Fourth Term

Again applying the recursive formula, we find \(a_4 = a_3 \, a_2 = 6 \times 3 = 18\).

Step 5: List the First Four Terms

Now that we have computed \(a_3\) and \(a_4\), the first four terms of the sequence are 2, 3, 6, and 18.

Step 6: Graph the Terms

Plot the points (1,2), (2,3), (3,6), and (4,18) on a graph. This will visually represent the sequence terms along the y-axis versus their position n along the x-axis.

Key Concepts

Sequence Graphing

Graphing the terms of a sequence helps us visualize how the sequence behaves over the initial terms. A sequence graph is a plot of discrete points, as each term relates only to its specific position within the sequence. To graph the sequence:

• Plot the first term as the point \( (1, a_1) \), here as \( (1, 2) \).
• Position the second term at \( (2, a_2) \), or \( (2, 3) \).
• For the third term, use \( (3, a_3) \), which is \( (3, 6) \).
• Finally, plot \( (4, a_4) \) as \( (4, 18) \).

Each point corresponds to a particular term \( n \) of the sequence.
In the graph, the x-axis represents the term number and the y-axis represents the term value. You will see how the terms grow, especially when the sequence terms increase quickly as shown in this example.

Sequence Terms

A sequence is built from a list of numbers, called terms, that follow a specific order. In a recursively defined sequence:

• Each term is calculated based on the previous terms in the sequence.
• Terms are denoted by \( a_n \), where \( n \) is the position number in the sequence.

For the sequence example we have, the first two terms are given:\[a_1 = 2, \ a_2 = 3\]Then, using the recursive relationship, we calculate:

• The third term: \( a_3 = a_2 \times a_1 = 3 \times 2 = 6 \)
• The fourth term: \( a_4 = a_3 \times a_2 = 6 \times 3 = 18 \)

Listing the first four terms, we get 2, 3, 6, and 18.
These terms show how the sequence evolves, and each term is essential for finding the next in the recursive pattern.

Recursive Formula

A recursive formula allows us to generate terms of a sequence by referring back to previous terms. This particular sequence is defined by the formula:\[a_n = a_{n-1} \times a_{n-2}\]Here, each term is the product of the two preceding terms. Important features of recursive formulas include:

• They offer a straightforward way to define sequences with complex relationships.
• Each term is uniquely determined once the initial terms are known.
• Recursive definitions can showcase exponential growth, as seen in our example.

Given \( a_1 = 2 \) and \( a_2 = 3 \), this rule creates a sequence starting with these numbers.
The recursive approach highlights the dynamic connection between sequence terms.
This simplicity facilitates understanding of potentially complex mathematical sequences.