Problem 19 - 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. \(a_{n}=3 a_{n-1}-1 ; a_{1}=2\)

Step-by-Step Solution

Answer

The first four terms are 2, 5, 14, and 41.

Step 1: Calculate the Second Term

The first term is given as \(a_1 = 2\). To find the second term \(a_2\), use the recursion formula: \[a_2 = 3 \cdot a_1 - 1 = 3 \cdot 2 - 1 = 6 - 1 = 5\]Thus, \(a_2 = 5\).

Step 2: Calculate the Third Term

Using the formula, calculate \(a_3\):\[a_3 = 3 \cdot a_2 - 1 = 3 \cdot 5 - 1 = 15 - 1 = 14\]So, \(a_3 = 14\).

Step 3: Calculate the Fourth Term

Now, calculate \(a_4\):\[a_4 = 3 \cdot a_3 - 1 = 3 \cdot 14 - 1 = 42 - 1 = 41\]Thus, \(a_4 = 41\).

Step 4: List the First Four Terms

The first four terms of the sequence are \(a_1 = 2\), \(a_2 = 5\), \(a_3 = 14\), and \(a_4 = 41\).

Step 5: Graph the Terms

Plot the points \((1, 2)\), \((2, 5)\), \((3, 14)\), and \((4, 41)\) on a graph. Connect these points with straight lines to visualize the sequence's growth.

Key Concepts

Sequence Terms

In the context of recursively defined sequences, sequence terms are individual elements in a sequence that follow a specific order. Each term is derived from the preceding one, according to a given rule. For this exercise, we start with the initial term, known as the first term, denoted as \(a_1\). It is provided as part of the sequence definition. Here, \(a_1 = 2\). Each subsequent term in the sequence is calculated using the recursion formula.

To find the second term \(a_2\), apply the recursion formula \(a_n = 3a_{n-1} - 1\). Substituting \(a_1\) gives us \(a_2 = 3 \times 2 - 1 = 5\). Similarly, calculate the third term \(a_3 = 3 \times 5 - 1 = 14\), and the fourth term \(a_4 = 3 \times 14 - 1 = 41\).

The sequence terms, therefore, emerge as a series: 2, 5, 14, and 41, each derived logically following the recursion rule. It's important for students to note how each term builds on the prior term, creating a pattern through repetition of the recursive process.

Graphing Sequences

Graphing a sequence involves plotting each of its terms on a coordinate system, which can help in visualizing its behavior or growth over time. For the given sequence, the terms are plotted as specific points on the graph, representing their position in the sequence along the x-axis and their value along the y-axis.

In this exercise, the first four terms are plotted as the points:

• (1, 2) for \(a_1=2\)
• (2, 5) for \(a_2=5\)
• (3, 14) for \(a_3=14\)
• (4, 41) for \(a_4=41\)

By connecting these points with straight lines, one can observe the growth pattern visually rather than just numerically. This can provide insights into whether the sequence is linear, exponential, or follows another growth type. With this sequence, you will notice a steep upwards trajectory, indicating rapid growth, characteristic of the multipliers used in its recursive formula.

Recursion Formula

A recursion formula is a rule that defines each term in a sequence using one or more of the preceding terms. This is a fundamental concept in recursively defined sequences, providing the mechanism to generate longer sequences from an initial term or terms.

For the sequence in this exercise, the recursion formula is \(a_n = 3a_{n-1} - 1\). This means each term \(a_n\) is calculated based on its predecessor \(a_{n-1}\). Recursion formulas often include both arithmetic operations and constant adjustments that shape the sequence's progression.

Understanding recursion formulas is crucial for students, as it emphasizes a step-by-step method of sequence generation. The careful application of this recursive rule allows one to predict future terms without recalculating from the very beginning. Once the pattern is understood, it can be extended indefinitely, aiding in both practical application and deeper mathematical exploration.

These formulas showcase mathematical elegance through simple repetition, offering a powerful tool for sequence development and analysis.