Problem 20 - 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. \(a_{n}=\frac{1}{2} a_{n-1} ; a_{1}=16\)

Step-by-Step Solution

Answer

First four terms are 16, 8, 4, and 2. Plot these against their indices on a graph.

Step 1: Understand the Given Recursive Formula

The sequence is defined as follows: for any term \(a_n\) it is half of the previous term \(a_{n-1}\). It starts with \(a_1 = 16\). This tells us how to calculate the next term by using the previous one.

Step 2: Calculate the First Term

The first term is provided directly: \(a_1 = 16\). This is the starting point of our sequence.

Step 3: Calculate the Second Term

Use the recursive formula to find \(a_2\). Since \(a_2 = \frac{1}{2} a_{1}\), substituting the value of \(a_1\) gives us \(a_2 = \frac{1}{2} \times 16 = 8\).

Step 4: Calculate the Third Term

Now, find \(a_3\) using the same formula: \(a_3 = \frac{1}{2} a_{2}\). Substituting \(a_2 = 8\) results in \(a_3 = \frac{1}{2} \times 8 = 4\).

Step 5: Calculate the Fourth Term

Determine \(a_4\) using the recursive formula: \(a_4 = \frac{1}{2} a_{3}\). With \(a_3 = 4\), we have \(a_4 = \frac{1}{2} \times 4 = 2\).

Step 6: Graph the Terms

Now that we have the first four terms of the sequence \(a_1 = 16\), \(a_2 = 8\), \(a_3 = 4\), and \(a_4 = 2\), plot these points on a graph with the term indices (1, 2, 3, and 4) as the x-axis and the term values as the y-axis. Connect them with lines to visualize how the sequence decreases.

Key Concepts

terms calculation

In recursive sequences, each term is derived from the previous one using a specific formula. For our exercise, the recursive sequence is defined by the rule: \[ a_{n} = \frac{1}{2} a_{n-1} \] with \( a_{1} = 16 \). This setup is quite straightforward. It means the value of any term is half the value of the preceding term. To find the initial terms, start with the first term given, and then apply the recursive rule thereafter:

• **First Term**: This is provided as \( a_{1} = 16 \).
• **Second Term**: Use the formula to get \( a_{2} = \frac{1}{2} \times 16 = 8 \).
• **Third Term**: Calculate next using \( a_{3} = \frac{1}{2} \times 8 = 4 \).
• **Fourth Term**: Finally, \( a_{4} = \frac{1}{2} \times 4 = 2 \).

This step-by-step approach helps build an understanding of how each term is reduced gradually through the recursive rule.

sequence graphing

Graphing a sequence can visually illustrate the behavior and direction in which the terms are progressing. To graph the terms of our sequence, place term indices on the x-axis and their corresponding values on the y-axis:

• **Point for First Term**: At (1, 16)
• **Point for Second Term**: At (2, 8)
• **Point for Third Term**: At (3, 4)
• **Point for Fourth Term**: At (4, 2)

After plotting these points, draw lines connecting them sequentially. This will form a straight descending line, illustrating the sequence's decrease over time. Using a graph is very helpful to understand how quickly or slowly a sequence drops, and, in our example, you can observe a consistent halving of the previous terms.

algebra problem-solving

To tackle any algebra-based sequence problems, recognizing patterns and relationships between terms is key. The recursive formula directly helps in identifying this relationship:\[ a_{n} = \frac{1}{2} a_{n-1} \]When solving algebra problems involving sequences, approach them with these principles:

• **Identify the Base**: Always start with what is given, typically the first term.
• **Understand the Pattern**: Look for how terms are manipulated to proceed to the next. Here, it's halving the term.
• **Apply Systematically**: Methodically apply the rule to find subsequent terms.

These steps turn a seemingly complex problem into manageable parts, where each term calculation builds directly off the last, enabling you to solve effectively and efficiently. Remember that consistent practice with this approach makes tackling recursive sequences more intuitive over time.