Problem 17 - 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. \(a_{n}=a_{n-1}+3 ; a_{1}=-3\)

Step-by-Step Solution

Answer

First four terms are -3, 0, 3, 6; plot them at (1,-3), (2,0), (3,3), (4,6).

Step 1: Understand the Recursive Definition

The given sequence is defined recursively, where the current term is calculated from the previous term. Specifically, each term in the sequence is obtained by adding 3 to the previous term. The sequence starts from the given first term: \( a_1 = -3 \).

Step 2: Calculate the First Term

According to the recursive rule, the first term is already provided: \( a_1 = -3 \). This serves as the base term for the sequence.

Step 3: Calculate the Second Term

Use the recursive formula to find the second term: \( a_2 = a_1 + 3 = -3 + 3 = 0 \).

Step 4: Calculate the Third Term

Continuing with the recursion, calculate the third term: \( a_3 = a_2 + 3 = 0 + 3 = 3 \).

Step 5: Calculate the Fourth Term

Using the recursive rule, find the fourth term: \( a_4 = a_3 + 3 = 3 + 3 = 6 \).

Step 6: List the Calculated Terms

The first four terms of the sequence are: \( a_1 = -3 \), \( a_2 = 0 \), \( a_3 = 3 \), \( a_4 = 6 \).

Step 7: Graph the Sequence

Plot these terms on a graph, treating the term number as the x-axis and the term value as the y-axis: (1, -3), (2, 0), (3, 3), (4, 6). Plot these points and connect them to visually represent the sequence.

Key Concepts

Sequence Graphing

Graphing a sequence provides a visual representation of how the terms change as you progress through the sequence. Understanding the pattern of a sequence is much easier when you can see it. For a recursively defined sequence, you graph each term by treating the term number as the x-coordinate and the term value as the y-coordinate. This approach allows you to see the relationship between sequential terms.

• Start by plotting each term using its position in the sequence for the x-axis values.
• The term values will be plotted on the y-axis.
• For our sequence: plot (1, -3), (2, 0), (3, 3), and (4, 6).
• Connect these points, typically with a straight line, to illustrate the sequence's progression.

Graphing helps reveal if a sequence increases, decreases, or alternates. In the case above, each point is progressively rising, indicating an arithmetic sequence with a positive constant difference between the terms.

Term Calculation

Term calculation in a recursively defined sequence involves generating subsequent terms based on a predefined rule. Using a recursive formula, each term depends directly on the value of its predecessor. Let's walk through calculating the first few terms of our sequence.

• Start with the given initial term, or base case, which for this sequence is \(a_1 = -3\).


• Apply the recursive formula to find the next term: \(a_n = a_{n-1} + 3\).


• Calculate the second term: \(a_2 = a_1 + 3 = -3 + 3 = 0\).
• Next, calculate the third term: \(a_3 = a_2 + 3 = 0 + 3 = 3\).
• Lastly, calculate the fourth term: \(a_4 = a_3 + 3 = 3 + 3 = 6\).

This step-by-step method helps build an understanding of how each term relates to the previous one, reinforcing the recursive nature of the sequence.

Recursive Formula

A recursive formula is a way to define a sequence by specifying how each term is related to the preceding ones. Unlike an explicit formula, which provides a direct means to calculate any term, recursive formulas rely on previous terms to find the next one.

• Recursive formulas are written with the form \(a_n = f(a_{n-1})\), where \(f\) is a function using the previous term.


• For our sequence, the recursive formula is \(a_n = a_{n-1} + 3\).
• Here, \(a_1 = -3\) is given, acting as our starting point or the known value.


• Each subsequent term uses this rule: take the last term and add 3, illustrating a simple arithmetic progression.

A recursive approach emphasizes understanding the relationship between terms and allows for constructing sequences effectively, though calculating specific terms necessitates knowledge of previous ones.