Problem 22 - 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. \(a_{n}=2 a_{n-1}+a_{n-2} ; a_{1}=0, a_{2}=1\)

Step-by-Step Solution

Answer

The first four terms are 0, 1, 2, and 5. Graph the points (1,0), (2,1), (3,2), (4,5).

Step 1: Understand the Sequence Definition

The sequence is defined recursively, which means each term depends on the previous terms. Specifically, the formula is \( a_{n} = 2a_{n-1} + a_{n-2} \) with initial conditions \( a_{1} = 0 \) and \( a_{2} = 1 \).

Step 2: Calculate the Third Term \( a_{3} \)

To find \( a_{3} \), substitute \( a_{1} = 0 \) and \( a_{2} = 1 \) into the recursive formula: \[a_{3} = 2a_{2} + a_{1} = 2 \times 1 + 0 = 2.\]

Step 3: Calculate the Fourth Term \( a_{4} \)

Use the terms \( a_{2} \) and \( a_{3} \) to find \( a_{4} \):\[a_{4} = 2a_{3} + a_{2} = 2 \times 2 + 1 = 5.\]

Step 4: List the First Four Terms

With \( a_{1} = 0 \), \( a_{2} = 1 \), \( a_{3} = 2 \), and \( a_{4} = 5 \), the first four terms of the sequence are 0, 1, 2, and 5.

Step 5: Graph the Calculated Terms

Plot the points representing the terms of the sequence on a graph with the x-axis representing the term number and the y-axis representing the term value. Points to plot: (1, 0), (2, 1), (3, 2), (4, 5). Connect these points to show the trend of the sequence.

Key Concepts

Sequence Graphing

Graphing a recursive sequence involves plotting its calculated terms on a graph to visually understand its behavior. For our sequence, we have determined the first four terms as 0, 1, 2, and 5. To graph these, do the following:

1. Place the term number (1, 2, 3, 4) on the x-axis.
2. Plot the corresponding term values (0, 1, 2, 5) on the y-axis.

This shows the relation between the sequence's order and its value.

By connecting these points with a line, the graph provides a visual trend of growth in the sequence. This visualization helps to anticipate the next terms and analyze the increase, which might be helpful to understand more complex behaviors in sequences.

Initial Conditions

Initial conditions are critical in defining a recursive sequence. In a sequence, these values kickstart the calculation of subsequent terms.

Let's consider the initial conditions in our sequence, which are given as:

• \( a_1 = 0 \)


• \( a_2 = 1 \)

These conditions were used to calculate successive terms like \( a_3 \) and \( a_4 \). Without these starting points, a recursive sequence cannot proceed because it relies on previous terms to define the next ones.

Think of it like building a house; if you don't have a foundation (initial conditions), you can't add the walls (subsequent terms). Initial conditions anchor the entire sequence, making them vital for sequence calculation.

Sequence Calculation

Calculating a recursively defined sequence might seem complex at first, but it follows a straightforward pattern. Start from the initial terms and use the recursive formula.The given sequence has the rule \( a_n = 2a_{n-1} + a_{n-2} \), which guides you to find each next term based on its predecessors.Here's the step-by-step for recalculating the first four terms:

• Calculate \( a_3 \): Insert \( a_1 = 0 \) and \( a_2 = 1 \) into the formula: \( a_3 = 2 \times 1 + 0 = 2 \).


• Calculate \( a_4 \): Now, use \( a_2 = 1 \) and \( a_3 = 2 \): \( a_4 = 2 \times 2 + 1 = 5 \).

Thus, our first four terms are \( 0, 1, 2, 5 \). The process is repetitive and follows from one calculation to the next, building the sequence term by term through recursive steps.