Problem 28 - 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. \(a_{n}=2 a_{n-1}^{2}+a_{n-2} ; a_{1}=2, a_{2}=1\)

Step-by-Step Solution

Answer

The first four terms are 2, 1, 4, and 33. Graph these terms as points on a graph.

Step 1: Understanding the Recursive Formula

The sequence is defined by the recurrence relation \( a_n = 2a_{n-1}^2 + a_{n-2} \). We start with \( a_1 = 2 \) and \( a_2 = 1 \). Our task is to find the next terms in the sequence using this formula.

Step 2: Calculating the Third Term

To find \(a_3\), use the formula with \(a_1 = 2\) and \(a_2 = 1\):\[a_3 = 2(1)^2 + 2 = 2 + 2 = 4.\]So, \(a_3 = 4\).

Step 3: Calculating the Fourth Term

Use the known values of \(a_2 = 1\) and \(a_3 = 4\) to calculate \(a_4\):\[a_4 = 2(4)^2 + 1 = 2 \times 16 + 1 = 32 + 1 = 33.\]Thus, \(a_4 = 33\).

Step 4: Review of First Four Terms

Now, we have the first four terms of the sequence: \(a_1 = 2\), \(a_2 = 1\), \(a_3 = 4\), and \(a_4 = 33\).

Step 5: Graphing the Terms

Plot the points \((1, 2)\), \((2, 1)\), \((3, 4)\), and \((4, 33)\) on a graph. Each \(x\)-coordinate corresponds to \(n\), and each \(y\)-coordinate corresponds to \(a_n\). This will visually represent the growth of the sequence.

Key Concepts

Recurrence Relation

In mathematics, a recurrence relation is an equation that defines a sequence based on preceding terms. It can be thought of as a set of instructions that tells you how to get from one term to the next. For this exercise, our sequence is defined by the recurrence relation:

• \(a_n = 2a_{n-1}^2 + a_{n-2}\)

This means to find any term \(a_n\), you take twice the square of the term before it, \(a_{n-1}\), and add the term two places before in the sequence, \(a_{n-2}\).
Recurrence relations can be challenging for beginners, as they're not a simple linear relationship like addition or multiplication by a constant number. Instead, they often involve nonlinear operations and multiple prior terms.
Understanding how they work is crucial for solving problems like this, where you're asked to find a set of initial terms or to express more general terms of a sequence. By starting with given initial terms, like \(a_1 = 2\) and \(a_2 = 1\), and calculating subsequent terms using the recurrence relation, you provide a path for completing the sequence.

Sequence Graphing

Sequence graphing is an excellent way to visualize how a series of numbers behaves as defined by a recurrence relation. In this exercise, once we found the first four terms, the next step was to plot them. Visualization helps grasp the changes and trends within the sequence at a glance.
When graphing sequences, each term is represented as a point, where the position on the x-axis corresponds to the term number (\(n\)), and the y-axis corresponds to the value of that term (\(a_n\)). For example:

• Plot the point \((1, 2)\) for \(a_1\)
• Plot the point \((2, 1)\) for \(a_2\)
• Plot the point \((3, 4)\) for \(a_3\)
• Plot the point \((4, 33)\) for \(a_4\)

This plotting shows not only the individual value of each term but also the growth or decline pattern, if any, within the sequence.
Graphing becomes particularly useful for recognizing patterns in sequences that might not be as apparent from just observing numbers. It is a tool that complements the algebraic representation, giving it a visual dimension.

Calculation of Terms

Calculating terms in a recursive sequence involves systematically using the recurrence relation, applying it step-by-step to determine specific values. This exercise demonstrates that process. We started with two known values, \(a_1 = 2\) and \(a_2 = 1\), and needed to find the next terms in the sequence.
To calculate \(a_3\), we inserted these known values into the recurrence relation:

• \(a_3 = 2(1)^2 + 2 = 2 + 2 = 4\)

Next, using \(a_2 = 1\) and the new \(a_3 = 4\) to calculate \(a_4\):

• \(a_4 = 2(4)^2 + 1 = 32 + 1 = 33\)

These calculations show how each term relies on the previous terms, illustrating the interconnected nature of recursive sequences.
This process can be extended to find additional terms, as needed. By repeatedly applying the recurrence relation, students can build as long a sequence as necessary. Understanding and practicing this concept is essential, especially when tackling more complex sequences or real-world applications where predictions or extended series calculations are required.