Problem 49 - College Algebra

Question

For the following exercises, graph the first five terms of the indicated sequence $$ a_{1}=2, a_{n}=\left(-a_{n-1}+1\right)^{2} $$

Step-by-Step Solution

Answer

The first five terms are: 2, 1, 0, 1, 0.

Step 1: Understand the sequence definition

The problem gives us the first term of the sequence, \(a_1 = 2\), and provides a recursive formula for the sequence: \(a_n = (-a_{n-1} + 1)^2\). This means that each term is determined by taking the previous term, subtracting 1 from it, multiplying the result by -1, and squaring it.

Step 2: Calculate the second term, \(a_2\)

Using the formula, calculate the second term:\[a_2 = (-a_1 + 1)^2 = (-(2) + 1)^2 = (-1)^2 = 1\\]Thus, \(a_2 = 1\).

Step 3: Calculate the third term, \(a_3\)

Using the formula, calculate the third term:\[a_3 = (-a_2 + 1)^2 = (-(1) + 1)^2 = (0)^2 = 0\\]Thus, \(a_3 = 0\).

Step 4: Calculate the fourth term, \(a_4\)

Using the formula, calculate the fourth term:\[a_4 = (-a_3 + 1)^2 = (-(0) + 1)^2 = (1)^2 = 1\\]Thus, \(a_4 = 1\).

Step 5: Calculate the fifth term, \(a_5\)

Using the formula, calculate the fifth term:\[a_5 = (-a_4 + 1)^2 = (-(1) + 1)^2 = (0)^2 = 0\\]Thus, \(a_5 = 0\).

Step 6: Plot the sequence on a graph

Plot the points corresponding to the terms of the sequence: (1,2), (2,1), (3,0), (4,1), (5,0). Each coordinate pair corresponds to the position (n) and value of \(a_n\). Notice the oscillating values.

Key Concepts

Graphing Sequences

Graphing sequences is a great way to visually understand how the terms are changing as you progress through the sequence. In our exercise, we were given a recursive formula that helps us determine each term based on the previous term. As we calculated the initial terms: 2, 1, 0, 1, 0, we can plot them on a graph to see the pattern. To graph a sequence:

• Identify each term's position on the x-axis, which represents the number of the term (e.g., 1st, 2nd, 3rd,...).
• Mark the value of the term on the y-axis.
• Plot the points for each term you have calculated.

Adding these points gives you a visual view, revealing how the sequence behaves over time. In our example, the graph shows an oscillating pattern, meaning the terms switch back and forth between values mainly around 1 and 0.

Algebraic Sequences

Algebraic sequences involve the use of algebraic rules or formulas to generate the terms of the sequence. The recursive formula given in our exercise is an example of such an algebraic rule. It uses algebraic operations: addition, subtraction, and squaring, to produce the terms. When dealing with recursive sequences:

• Ensure you understand the formula given, as it usually relies on previous terms.
• Carefully apply the operations as written to get the next term.
• Keep track of each term, as each serves as a building block for the next term.

Algebraic sequences can range from simple to complex, but practicing with recursive examples like this can help build your understanding of mathematical relationships.

Sequence Terms

Sequence terms are the individual elements that make up the sequence. Each term is crucial in understanding the overall behavior of the sequence. In our example, we started with an initial term of 2, and each subsequent term was derived from the previous one using a specific formula. Key points about sequence terms:

• The first term is often provided, as it is needed to start calculating the subsequent terms.
• Each term represents a point in the progression and is numbered accordingly, like the 1st term, 2nd term, and so on.
• Terms are calculated systematically based on the sequence's rule.

Understanding sequence terms simplifies the overall concept, allowing you to predict and graph future terms effectively.

Oscillating Sequences

Oscillating sequences are those that exhibit a back-and-forth pattern, alternating between certain values. In our problem, the sequence terms alternate primarily between 1 and 0 after the initial term of 2. Characteristics of oscillating sequences include:

• A noticeable pattern where terms repeat or alternate in a specific manner.
• Often seen in graphs as a wave-like pattern, reflecting the points going up and down.
• Recognizing this pattern helps in predicting future terms and understanding the sequence behavior.

Oscillating sequences are common in mathematical functions and real-world scenarios, making them an important concept to grasp when studying sequences.