Problem 18 - 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. \(a_{n}=2 a_{n-1}+1 ; a_{1}=1\)

Step-by-Step Solution

Answer

The first four terms are 1, 3, 7, and 15.

Step 1: Identify the First Term

The sequence is given as recursive, with the first term provided: \(a_1 = 1\). This means that the first term \(a_1\) is 1.

Step 2: Compute the Second Term

To find the second term \(a_2\), use the recursive formula provided: \(a_n = 2a_{n-1} + 1\). Plug in \(a_1 = 1\): \[a_2 = 2 \times 1 + 1 = 3\] Therefore, \(a_2 = 3\).

Step 3: Compute the Third Term

Now calculate the third term \(a_3\) using the recursive formula and \(a_2 = 3\): \[a_3 = 2 \times 3 + 1 = 7\] Thus, \(a_3 = 7\).

Step 4: Compute the Fourth Term

Find \(a_4\) using the formula and \(a_3 = 7\): \[a_4 = 2 \times 7 + 1 = 15\] This gives \(a_4 = 15\).

Step 5: Graph the Terms

For the graph, plot the points corresponding to the four terms: (1, 1), (2, 3), (3, 7), and (4, 15). Use a graphing tool or graph paper to plot and connect these points to visualize the sequence.

Key Concepts

Algebra

Recursive sequences in algebra are fascinating because they help us understand how terms in a sequence relate to each other through an operation. A recursive formula defines each term using the previous term, making it a step-by-step process to find subsequent terms. In our exercise, the sequence is defined by the formula \[ a_{n} = 2a_{n-1} + 1 \] and an initial condition \( a_1 = 1 \).

• **Initial Condition**: This sets the starting point of the sequence.
• **Recursive Formula**: Provides a rule to find the next term.

Let's break down how we use these components:- The first term \( a_1 \) is simply 1, according to the given initial condition.
- To find the second term \( a_2 \), we substitute the first term into the formula: \[ a_2 = 2 imes a_1 + 1 = 3 \]- For the third term \( a_3 \), we use the second term: \[ a_3 = 2 imes a_2 + 1 = 7 \]- The process repeats to find \( a_4 \): \[ a_4 = 2 imes a_3 + 1 = 15 \]This expansion relies on algebra skills to substitute and solve for the next terms accurately.

Mathematical Graphing

Graphing a recursive sequence helps visualize the growth pattern of the sequence over time. When you plot the terms of the sequence, it turns the abstract numbers into a visual hopscotch of points that bring clarity and insight.

• **Plotting Points**: We represent each term as a point on a graph. In our sequence, these points are (1, 1), (2, 3), (3, 7), and (4, 15).
• **Connecting Dots**: While connecting these dots directly isn't always meaningful as with continuous data, observing their trajectory gives insight into the sequence's behavior, like growth patterns.

Each point on the graph corresponds to a term in the sequence:- **First Point**: (1, 1) for term \( a_1 \)- **Second Point**: (2, 3) for term \( a_2 \)- **Third Point**: (3, 7) for term \( a_3 \)- **Fourth Point**: (4, 15) for term \( a_4 \)By graphing these, you can quickly see that our sequence grows exponentially, a concept seen visually through the steep, upward curve of the plotted points.

Step-by-step Solutions

Step-by-step solutions are crucial in building your confidence and competence in handling recursive sequences. They break down complex processes into manageable tasks that follow logically connected steps.

• **Simplification**: Each step focuses on one part of the task, like finding a term in the sequence, before moving on to the next.
• **Clarity**: Seeing each term in relation to the previous one helps solidify understanding of recursive rules.

Here's how each step aids problem-solving: 1. **Step 1**: Establish the initial term from the given data. It sets the groundwork for the sequence.
2. **Step 2 - 4**: Calculate each subsequent term using the recursive rule. Move sequentially to prevent errors.
3. **Step 5**: The final step involves graphing the terms to visualize your calculations, cementing your understanding of both algebra and graphing simultaneously. This approach is invaluable in learning mathematics because it teaches you how not to rush but instead to understand the problem deeply.