Problem 23 - 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. \(a_{n}=a_{n-1}^{2} ; a_{1}=2\)

Step-by-Step Solution

Answer

The first four terms are 2, 4, 16, and 256; graph these terms accordingly.

Step 1: Understand the Recursive Formula

The sequence is defined recursively, which means each term is based on the previous one. The given recursive formula is \(a_{n}=a_{n-1}^{2}\) and the initial term \(a_1 = 2\). This indicates that \(a_2\) will be calculated from \(a_1\), \(a_3\) from \(a_2\), and so on.

Step 2: Calculate the Second Term

Using the recursive formula, substitute \(a_1\) into \(a_2 = a_1^2\). Replace \(a_1\) with 2, so \(a_2 = 2^2 = 4\).

Step 3: Calculate the Third Term

Substitute \(a_2\) into \(a_3 = a_2^2\). So, \(a_3 = 4^2 = 16\).

Step 4: Calculate the Fourth Term

Substitute \(a_3\) into \(a_4 = a_3^2\). Compute to find \(a_4 = 16^2 = 256\).

Step 5: List the First Four Terms

By following the above calculations, the first four terms of the sequence are: \(a_1 = 2\), \(a_2 = 4\), \(a_3 = 16\), \(a_4 = 256\).

Step 6: Graph the Terms

Plot the points on a graph with the x-axis showing the sequence number (n) and the y-axis indicating the term value \(a_n\). Plot points (1,2), (2,4), (3,16), and (4,256) to visualize the sequence.

Key Concepts

Sequence Terms

A sequence is essentially a list of numbers called terms, arranged in a particular order. In a recursively defined sequence, each term is generated from the previous one using a specific rule. In this case, we have the formula \(a_{n} = a_{n-1}^{2}\) with an initial term \(a_1 = 2\). This means that each term is the square of the term immediately before it.
To find the first few terms, you start with the initial term and systematically apply the rule:

• Start with \(a_1 = 2\).
• For the second term, square the first term: \(a_2 = 2^2 = 4\).
• For the third term, square the second term: \(a_3 = 4^2 = 16\).
• For the fourth term, square the third term: \(a_4 = 16^2 = 256\).

So, the terms are \(2, 4, 16,\) and \(256\). You can continue this process to find more terms, applying the recursive relationship each time.

Recursive Formula

A recursive formula defines a sequence by expressing terms as a function of their predecessor(s). It allows you to build the sequence by starting with one or more initial terms. With the formula \(a_{n} = a_{n-1}^{2}\), and \(a_1 = 2\), you derive the subsequent terms systematically.
The recursive formula is beneficial because:

• It simplifies the process of finding terms after the initial one.
• It expresses a complex sequence with simple, easy-to-follow steps.
• Computation is based on previously found terms, which can help identify patterns or specific properties of the sequence.

To use a recursive formula effectively, understand the initial term and apply the recursive step precisely. Initial terms act as the starting point, and missing these would make it impossible to calculate any further terms. This approach is both methodical and creative, allowing you to explore the sequence as it evolves.

Graphing Sequences

Graphing a sequence provides a visual representation of how the sequence behaves. This can help you quickly observe patterns or the nature of growth or decay. For the given sequence, we need to plot the first four terms: \(a_1 = 2\), \(a_2 = 4\), \(a_3 = 16\), and \(a_4 = 256\).
To graph these:

• Use the x-axis for the term number \(n\), and the y-axis for the term values \(a_n\).
• Place points on the graph at (1,2), (2,4), (3,16), and (4,256).
• Notice the shape of the graph. In this case, it should show exponential growth due to the squaring operation of each term from the last.

Visualizing sequences through graphs aids in understanding their trajectory and growth rates. Graphs are especially useful when analyzing more complex sequences, making it easier to detect cyclical patterns or other behaviors.