Problem 24 - Algebra for College Students

Question

Write the first four terms of the infinite sequence whose nth term is given. \(c_{n}=(-1)^{n}(2 n-1)^{2}\)

Step-by-Step Solution

Answer

The first four terms are: -1, 9, -25, and 49.

Step 1 - Understand the formula

The formula for the nth term of the sequence is given by \(c_{n}=(-1)^{n}(2 n-1)^{2}\). Evaluate this formula for the first four values of \(n\).

Step 2 - Calculate the 1st term

Set \(n = 1\). Substitute it into the formula: \(c_{1}=(-1)^{1}(2 \times 1-1)^{2} = (-1) \times 1^2 = -1\). So, the first term is \(-1\).

Step 3 - Calculate the 2nd term

Set \(n = 2\). Substitute it into the formula: \(c_{2}=(-1)^{2}(2 \times 2-1)^{2} = (1) \times 3^2 = 9\). So, the second term is \(9\).

Step 4 - Calculate the 3rd term

Set \(n = 3\). Substitute it into the formula: \(c_{3}=(-1)^{3}(2 \times 3-1)^{2} = (-1) \times 5^2 = -25\). So, the third term is \(-25\).

Step 5 - Calculate the 4th term

Set \(n = 4\). Substitute it into the formula: \(c_{4}=(-1)^{4}(2 \times 4-1)^{2} = (1) \times 7^2 = 49\). So, the fourth term is \(49\).

Key Concepts

nth term calculation

In order to fully understand the concept of sequence operations, let’s dive deeper into the calculation of the nth term. The nth term of a sequence is a formula that allows you to find any term in the sequence without having to list all prior terms.

In the given exercise, the nth term is defined by the formula: \( c_{n}=(-1)^{n}(2n-1)^{2}\).

To calculate the nth term:

• Identify the term number you want to find (n = 1, 2, 3, ...).
• Substitute the term number into the formula.
• Simplify the expression to find the value of the term.

As shown in the step-by-step solution, this systematic approach makes it easy to find specific terms in the sequence.

infinite sequences

Infinite sequences are an essential concept in algebra and higher mathematics. An infinite sequence is a list of numbers that continues indefinitely. They can be defined by a formula, as demonstrated in the exercise.

For example, the sequence given by \(c_{n}=(-1)^{n}(2n-1)^{2}\) is infinite because there is no endpoint to the values of n. The sequence will continue generating terms as long as we keep inputting values of n.

Some properties of infinite sequences include:

• They do not have a last term.
• They can follow a specific pattern or rule.
• They are often used in various mathematical analyses and applications.

Understanding infinite sequences helps in exploring deeper mathematical theories and solving complex problems.

evaluating sequences

Evaluating sequences involves finding specific terms in the sequence or determining properties of the sequence like its limit or sum (if the sequence is summable).

To evaluate a sequence effectively:

• Use the given nth term formula to find specific terms.
• Notice the pattern of the sequence (e.g., alternating signs or increasing values).
• Understand how the sequence behaves as n grows larger (for example, does it tend to infinity or oscillate between values?).

In the exercise, specific terms are evaluated using the formula \(c_{n}=(-1)^{n}(2n-1)^{2}\). By substituting values of n = 1, 2, 3, and 4, we find the first four terms as -1, 9, -25, and 49 respectively.

This approach helps in comprehending how sequences are structured and provides insights into their broader mathematical applications.