Problem 26 - 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. \(a_{n}=3 a_{n-1}^{2} ; a_{1}=2\)

Step-by-Step Solution

Answer

The first four terms are 2, 12, 432, and 559872.

Step 1: Understand the Recursive Formula

The recursive sequence is defined by \( a_{n} = 3a_{n-1}^2 \) with the initial condition \( a_1 = 2 \). This means each term is computed based on the previous term.

Step 2: Compute the First Term

Since we are given that \( a_1 = 2 \), the first term of the sequence is \( 2 \).

Step 3: Compute the Second Term

Use the recursive formula to find the second term: \( a_2 = 3 \times (a_1)^2 = 3 \times 2^2 = 3 \times 4 = 12 \).

Step 4: Compute the Third Term

Use the recursive formula to find the third term: \( a_3 = 3 \times (a_2)^2 = 3 \times 12^2 = 3 \times 144 = 432 \).

Step 5: Compute the Fourth Term

Use the recursive formula to find the fourth term: \( a_4 = 3 \times (a_3)^2 = 3 \times 432^2 = 3 \times 186624 = 559872 \).

Step 6: List the First Four Terms

The first four terms of the sequence are \( a_1 = 2 \), \( a_2 = 12 \), \( a_3 = 432 \), and \( a_4 = 559872 \).

Step 7: Graph the Terms

Create a graph with the x-axis representing the term index \( n \) and the y-axis representing the term value \( a_n \). Plot the points (1, 2), (2, 12), (3, 432), and (4, 559872). You'll notice the graph increases very steeply.

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In the context of recursive sequences, algebra helps us express terms and relationships in a sequence with formulas. When we examine the sequence \( a_n = 3a_{n-1}^2 \), we are essentially using algebra to set up a pattern that describes how each term in the sequence is derived from the previous one. This requires basic algebraic manipulation like squaring terms and multiplying by constants.

• Expression: A mathematical phrase that can contain numbers, variables, and operation symbols.
• Term: Each element in a sequence or equation, like \( a_1, a_2, a_3 \), etc.
• Formula: An equation that calculates a new term based on known information.

Understanding these algebraic fundamentals allows us to delve deeper into sequences and their behavior.

Sequence Graphing

Sequence graphing involves representing terms of a sequence as points on a coordinate system. In our exercise, we graphed the sequence defined by \( a_n = 3a_{n-1}^2 \) and started with \( a_1 = 2 \). When we plot the terms \((1, 2), (2, 12), (3, 432), \) and \((4, 559872)\), each point corresponds to a term index \( n \) and its value \( a_n \).Graphing is insightful because:

• Visually shows the pattern: As seen, terms increase rapidly, revealing exponential growth.
• Identify trends: For recursive sequences, graphing can highlight how dramatically terms can change.
• Scientific tools: Graphs are essential for predicting future terms or understanding complex sequences.

While the sequence initially seems simple, graphing exhibits its inherent complexity and rapid increase.

Term Calculation

Term calculation in a recursive sequence involves determining each term based on the preceding one, using a given recursive formula. Let’s break down the calculation of terms in our exercise sequence.1. **Initial Term**: The sequence starts with an initial term, \( a_1 = 2 \).2. **Use of Recursive Formula**: For each subsequent term, we plug the previous term into the equation \( a_n = 3a_{n-1}^2 \): - Second term: \( a_2 = 3 \times a_1^2 = 3 \times 4 = 12 \) - Third term: \( a_3 = 3 \times a_2^2 = 3 \times 144 = 432 \) - Fourth term: \( a_4 = 3 \times a_3^2 = 3 \times 186624 = 559872 \)These steps underscore how each term depends strictly on its predecessor, showing the essence of recursive sequences.

Recursion

Recursion refers to a process where a function, or in our case, a sequence, is defined in terms of itself. This concept is central to defining recursive sequences like \( a_n = 3a_{n-1}^2 \). Understanding recursion involves grasping:

• Base case (starting point): Here, \( a_1 = 2 \) provides the initial term of our sequence.
• Recursive case (formula): Every new term builds on the result of its predecessor through the formula \( a_n = 3a_{n-1}^2 \).

Recursion is potent because it systematically breaks down complex problems into simpler ones. In sequences, it allows us to form extensive chains of values from just a simple starting condition and a rule that shows how each term evolves, illustrating the power of defining terms repetitively.