Problem 65 - Elementary Algebra

Question

What are consecutive odd integers? Name three consecutive odd integers between 50 and 60 .

Step-by-Step Solution

Answer

Three consecutive odd integers between 50 and 60 are 51, 53, and 55.

Step 1: Understand Consecutive Odd Integers

Consecutive odd integers are odd numbers that follow one another in sequence. For example, 1, 3, and 5 are consecutive odd integers.

Step 2: Identify the Range

The given range is between 50 and 60. Identify the odd integers within this range.

Step 3: List Odd Integers Within the Range

The odd integers between 50 and 60 are 51, 53, 55, 57, and 59.

Step 4: Choose Three Consecutive Odd Integers

From the list of odd integers, choose any set of three consecutive numbers. For example, 51, 53, and 55, or 53, 55, and 57, etc.

Key Concepts

Odd Numbers

Odd numbers are integers that cannot be divided evenly by 2. In other words, when you divide an odd number by 2, there will always be a remainder of 1. Examples of odd numbers include 1, 3, 5, and 7. These numbers differ from even numbers, which are multiples of 2, such as 2, 4, 6, and 8.
Odd numbers play a crucial role in mathematics and real-life scenarios. Understanding the properties of odd numbers helps in solving various problems.
Here are a few properties of odd numbers:

• Adding two odd numbers gives an even number. For example, 3 + 5 = 8.
• Multiplying two odd numbers results in an odd number. For example, 3 * 5 = 15.
• Odd numbers alternate with even numbers on a number line (e.g., 1, 2, 3, 4,...).

Knowing these properties can help you identify and work with odd numbers more effectively.

Integer Sequences

An integer sequence is an ordered list of integers. These sequences can follow different patterns, such as arithmetic sequences, geometric sequences, or more complex patterns. In the case of consecutive odd integers, the pattern is an arithmetic sequence, where each term increases by 2 from the previous term.
For example, the sequence of consecutive odd integers starting from 1 is 1, 3, 5, 7, and so on. Each number in the sequence is 2 more than the previous one.
Understanding integer sequences helps in recognizing patterns and making predictions. Some important characteristics of integer sequences include:

• The first term is often called the starting point.
• The difference between consecutive terms in an arithmetic sequence is called the common difference.
• In the context of odd integers, the common difference is always 2.

By recognizing the pattern in integer sequences, you can solve various types of mathematical problems efficiently.

Number Range Identification

Number range identification involves finding and listing numbers within a specified range that meet certain conditions. In the given exercise, the range is between 50 and 60, and we are looking for odd integers within this range.
To identify numbers within a range, follow these steps:

• Understand the boundaries of the range (in this case, 50 and 60).
• Determine the criteria for selecting numbers (here, the criterion is that the numbers must be odd).
• List the numbers that meet the criteria within the given range.

Let’s apply this to the exercise. The numbers between 50 and 60 are 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, and 60. To find the odd numbers in this range, we can simply list out every second number starting from 51:

• 51
• 53
• 55
• 57
• 59

Out of these, you can easily choose sets of three consecutive odd integers as required. Identifying ranges and criteria helps simplify the process of filtering and selecting the appropriate numbers.