Problem 91 - Intermediate Algebra

Question

Answer the question with an algebraic expression. If \(n\) represents an odd integer, what represents the next larger odd integer?

Step-by-Step Solution

Answer

The next odd integer is \(n + 2\).

Step 1: Recognize the Nature of Odd Integers

An odd integer can be represented in the form \(2k + 1\), where \(k\) is an integer. Hence, \(n = 2k + 1\), since \(n\) is an odd integer.

Step 2: Determine the Next Odd Integer

To find the next odd integer after \(n = 2k + 1\), add 2 to \(n\). This is because the difference between consecutive odd integers is always 2.

Step 3: Write the Expression for the Next Odd Integer

Add 2 to the expression for \(n\): \(n + 2 = (2k + 1) + 2 = 2k + 3\). Thus, the next odd integer is represented by \(n + 2 = 2k + 3\).

Key Concepts

Odd Integer

Odd integers are whole numbers that cannot be evenly divided by 2. That means when you divide an odd number by 2, you will have a remainder of 1. Examples of odd integers include 1, 3, 5, 7, and 9. In mathematical terms, odd integers can be represented using the formula \(2k + 1\), where \(k\) is any integer. This formula guarantees that the number will be odd because:

• Multiplying any integer \(k\) by 2 results in an even number, \(2k\).
• Adding 1 to an even number changes it to an odd one.

Understanding the basic representation of odd integers using \(2k + 1\) makes it simpler to engage in algebraic operations involving odd numbers.

Consecutive Integers

Consecutive integers are numbers that follow each other in the natural order, without any numbers missing between them. For example, if you have 1, 2, and 3, these numbers are consecutive integers. When dealing with odd integers, the concept remains the same, but it involves skipping even numbers.Let's look at two consecutive odd integers, say 3 and 5:

• The difference between each pair of consecutive odd integers is always 2.
• If \(n\) represents an odd integer, then \(n + 2\) will represent the next odd integer.

This pattern allows you to form sequences of odd integers and makes algebraic manipulations easier, as you know that every consecutive odd integer pair will differ by exactly 2.

Mathematical Representation

In algebra, representing mathematical ideas and patterns through expressions is crucial for solving problems efficiently. When working with integers, especially odd and consecutive integers, having a formula helps. For odd integers, as established, the generic formula is \(2k + 1\). To find the next odd integer after a given one, you add 2. Thus, if you have an odd integer represented by \(n = 2k + 1\), the next one is \(n + 2\), which simplifies to \(2k + 3\). The process is straightforward:

• Identify the integer using \(2k + 1\).
• Add 2 to this expression to find the next integer.

Using mathematical expressions allows you to predict patterns quickly without manual counting, ensuring accuracy in solving linear patterns and creating algebraic equations.