Problem 4 - Prealgebra

Question

Find the next term in list. \(12,17,22,27,32, \dots\)

Step-by-Step Solution

Answer

The next term is 37.

Step 1: Identify the Pattern

First, observe the difference between consecutive terms in the sequence: - From 12 to 17: Difference is 5 - From 17 to 22: Difference is 5 - From 22 to 27: Difference is 5 - From 27 to 32: Difference is 5 It seems that each term increases by 5.

Step 2: Apply the Pattern to Find the Next Term

To find the next term in the sequence, add the common difference of 5 to the last term in the sequence. The last term given is 32. So, the next term is: \[32 + 5 = 37\]

Key Concepts

Understanding Sequence Patterns

A sequence pattern refers to a repeated or regular order of numbers or elements that follow a specific rule. In arithmetic sequences, this pattern is identified by examining the regular alteration of each term in the sequence. Here, each step from the original number leads us predictably to the next number. Observing patterns in sequences helps us clearly visualize how sequences evolve.

• The sequence you've been given is: 12, 17, 22, 27, 32, ...
• In this list, the pattern to look out for is the difference between each number.

To identify the pattern, start by calculating the difference between consecutive terms. In this example:

• First, from 12 to 17, the difference is 5
• From 17 to 22, the difference is also 5
• This consistent difference continues through the series.

By understanding sequence patterns, particularly in arithmetic sequences, determining the next term becomes an intuitive task.

The Role of Common Difference

The common difference in an arithmetic sequence is the fixed number added to each term to get to the next term. For any arithmetic sequence, this difference is constant throughout the series.

• Having a common difference makes the sequence predictable and is crucial for finding subsequent terms.
• In our sequence example, the common difference is 5, as determined by the difference between each pair of terms.

To solve for the next term in an arithmetic sequence, simply add the common difference to the latest term provided. In this exercise, after the term 32, the next term is derived by adding 5. This gives 37 as the following number in the sequence.
Identifying and using the common difference allows for efficient predictions, even in sequences that extend indefinitely.

Exploring Mathematical Progression

A mathematical progression, specifically an arithmetic progression in this context, refers to a sequence where each term increases by the same amount, known as the common difference. This kind of progression is defined by linear growth, where terms move forward or backward in a predictable pattern.

• Arithmetic progressions provide a foundation for recognizing patterns in mathematics and real-world events where regular intervals are present.
• Such progressions are not only used to solve exercises but can also be applied to various fields like financial planning, predicting outcomes, and scheduling tasks.

To illustrate, consider the sequence 12, 17, 22, 27, 32, which we previously discussed. It demonstrates a simple arithmetic progression with a common difference of 5. The progression is used to predict the future or missing terms efficiently. By understanding mathematical progression, you gain broader insights into how structured numeric growth is formulated and analyzed.