Problem 1 - Algebra 2

Question

Describe each pattern formed. Find the next three terms. $$ 80,77,74,71,68, \dots $$

Step-by-Step Solution

Answer

The pattern is a constant decrease of 3. The next three terms are 65, 62, 59.

Step 1: Identify the pattern

The sequence presented: 80, 77, 74, 71, 68 shows a constant decrease of 3. This means each term is 3 less than the previous one.

Step 2: Apply the pattern to find the next term

To find the next term, subtract 3 from the last given term (68). So, \(68-3=65\). Therefore, the next term is 65.

Step 3: Repeat to find the following terms

Repeat the process to find the next two terms: \(65-3=62\) and \(62-3=59\). So, the following terms are 62 and 59.

Key Concepts

Patterns

Recognizing patterns is an essential part of understanding sequences in mathematics. A pattern is a repeated or recurring sequence that allows predictions about future numbers. In arithmetic sequences, the difference between consecutive terms remains constant. This constant difference is often referred to as the "common difference". Identifying this common difference is the first step in understanding the pattern of the sequence. For example, in the sequence provided: 80, 77, 74, 71, 68, we see that each term is obtained by subtracting 3 from the previous one. Thus, the pattern here is a constant decrease of 3.
To identify the pattern in such sequences, look at the progression of the numbers and pinpoint any consistent shift from one term to the next. This recognition will guide you in predicting the sequence's continuation.
Recognizing patterns not only helps in forecasting the next terms but also in understanding the underlying mathematical structure, which is crucial for solving more complex problems.

Sequences

Sequences are ordered lists of numbers that follow a specific rule or formula. They can be finite (having a limited number of terms) or infinite (continuing indefinitely). Each number in a sequence is called a 'term'. Understanding sequences involves finding the rule governing the formation of numbers.
In arithmetic sequences, like the one in our exercise, each term is derived from the previous one by adding or subtracting a constant value, known as the common difference. For example, the sequence 80, 77, 74, 71, 68 decreases by 3 each time, showing its arithmetic nature.
To fully grasp sequences, it's important to understand the notations and terms used. The nth term of an arithmetic sequence can be calculated using the formula: \[ a_n = a_1 + (n-1) imes d \] where \(a_n\) is the nth term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference. This formula allows you to find any term in the sequence without listing all previous terms.

Linear Patterns

Linear patterns are a type of pattern where the change between terms is constant. These patterns are represented as straight lines when plotted on a graph, which is a key characteristic of linear relationships. In the context of arithmetic sequences, linear patterns are evident through their uniform step as seen with the common difference.
For the provided sequence: 80, 77, 74, 71, 68, the linear pattern is characterized by the steady decrease of 3. This regularity aligns with their representation in a linear graph, where each term is a point that connects evenly to form a straight line.

• Each term decreases by the same amount (common difference of -3).
• The rule governing the sequence is linear, meaning that it changes at a constant rate.

Understanding linear patterns helps in predicting further terms in the sequence without exhaustive calculations, and it also provides insights into solving algebraic equations and other advanced mathematical concepts involving linear relationships.