Problem 4 - Algebra 2

Question

Describe each pattern formed. Find the next three terms. $$ 1,4,7,10,13, \dots $$

Step-by-Step Solution

Answer

The next three terms for the sequence \(1, 4, 7, 10, 13, \dots\) are 16, 19, and 22.

Step 1: Identify the pattern

Look at the differences between the terms in the sequence. For this sequence: \(1, 4, 7, 10, 13\), the numbers are increasing as follows: \(4-1=3\), \(7-4=3\), \(10-7=3\), and \(13-10=3\). It can be seen that the difference between consecutive terms is a constant of 3, this is an arithmetic sequence.

Step 2: Find the next terms using the pattern

Following the pattern identified, that is, adding 3 to the last term, the next term would be \(13+3=16\). Similarity, the term following that would be \(16+3=19\) and the term after that would be \(19+3=22\).

Key Concepts

Patterns in Sequences

Understanding patterns in sequences is the key to recognizing and forming different types of sequences. In an arithmetic sequence, like the one provided in the exercise, each term is obtained by adding a constant value, called the common difference, to the preceding term. This constant progression results in a linear sequence of numbers.
The importance of recognizing these patterns lies in using them to predict future terms without having to manually add each time. In the provided sequence:

• Start with 1
• Add 3 repeatedly to get the next terms

This repetitive addition of a fixed number helps in quickly generating the sequence and understanding that it's indeed an arithmetic progression.

Term Identification

Term identification in a sequence involves recognizing both the position and the value of each term. This is crucial for understanding how sequences are structured. In an arithmetic sequence, the terms follow a well-defined pattern, making it easy to pinpoint any term given its position.
Let's look at the sequence provided:

• Start with 1 (first term)
• The second term is calculated as 1 + 3 = 4
• The third term 4 + 3 = 7, and so on

The position of each term corresponds to an arithmetic progression based on the initial term and the common difference. By determining this, one can find any term in the sequence using the formula for the nth term: \[a_n = a_1 + (n-1) imes d\]where \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.

Sequence Analysis

Analyzing sequences allows us to understand their structure and derive meaningful conclusions, such as predicting future terms or determining the overall pattern. In this case, the sequence follows an arithmetic pattern where each subsequent term is obtained by adding a constant value to the previous term.
Through sequence analysis, we break down these components:

• Observe that each term is separated by a common difference of 3
• The sequence begins at 1 and progresses by 3: 1, 4, 7, 10, 13,...
• Future terms can be predicted by continuing the pattern

By analyzing this sequence, we can easily calculate future terms such as 16, 19, and 22, continuing beyond just the given terms. This understanding is not only useful for math problems but also applies in various fields where predicting trends is essential.