Problem 91 - Algebra for College Students

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 larger odd integer is represented by \( n + 2 \).

Step 1: Understand the Problem

We are given that \( n \) is an odd integer and need to find an expression for the next larger odd integer.

Step 2: Define an Odd Integer

An odd integer can be written in the form \( n = 2k + 1 \), where \( k \) is an integer. This form ensures that the integer is odd since it is one more than an even number \( 2k \).

Step 3: Find the Next Odd Integer

To find the next odd integer after \( n \), we add 2 to it. This means the next odd integer after \( n = 2k + 1 \) is \( n + 2 = 2k + 3 \).

Step 4: Generalize the Expression

Since we defined \( n \) as an odd integer \( 2k + 1 \), the expression for the next larger odd integer is simply \( n + 2 \).

Key Concepts

Odd Integers

Odd integers are numbers that cannot be evenly divided by 2. This means that when you divide an odd integer by 2, there will always be a remainder of 1. You can recognize odd integers by looking at the last digit. Numbers ending in 1, 3, 5, 7, or 9 are odd. For example, 3, 15, 27, and 59 are all odd integers.

In algebra, an odd integer can be represented as \( n = 2k + 1 \), where \( k \) is an integer. This formula works because \( 2k \) is always even, and adding 1 to an even number results in an odd number.

Understanding odd integers is fundamental in mathematics as it helps in identifying patterns and sequences, especially when dealing with number series or algebraic expressions.

Next Integer

When talking about the 'next integer' in relation to any given integer, it simply means the integer that comes directly after it. Specifically for odd integers, finding the next odd integer involves a small calculation.

For instance, if you have an odd integer represented by \( n = 2k + 1 \), you can find the next odd integer by adding 2. So, the next odd integer becomes \( n + 2 = 2k + 3 \).

This concept is useful in number theory and helps solve problems where sequences and predictions of integer patterns are required. After every odd integer, another odd one follows after skipping an even number. This is why adding 2 makes sense.

Integer Sequences

An integer sequence is an ordered list of integers. These sequences can follow various patterns or rules. One common type is an arithmetic sequence, where each number is a fixed amount apart. This is very applicable when discussing odd integers.

For example, if we start with an odd integer such as 1, and continually add 2, we get a sequence of odd integers: 1, 3, 5, 7, and so on. This clearly forms an arithmetic sequence with a common difference of 2.

In mathematics, understanding integer sequences helps in predicting future numbers, solving equations, and analyzing patterns. Recognizing these sequences equips you with the tools to solve a wide range of algebraic and geometric problems easily.