Problem 47 - Prealgebra

Question

What is the next term in \(67,62,57,52,47, \ldots ? \text { ( Lesson } 1-1)\)

Step-by-Step Solution

Answer

The next term is 42.

Step 1: Identify the Pattern

To solve this problem, we first need to determine the pattern in the sequence of numbers. Let's look at the difference between consecutive terms. The sequence given is: 67, 62, 57, 52, 47.

Step 2: Calculate Differences

Subtract consecutive terms to find the difference: - Between 67 and 62: 67 - 62 = 5 - Between 62 and 57: 62 - 57 = 5 - Between 57 and 52: 57 - 52 = 5 - Between 52 and 47: 52 - 47 = 5. The difference between consecutive terms is consistently 5.

Step 3: Determine the Pattern Type

Since the difference between each consecutive term is constant (5), the pattern in the sequence is an arithmetic sequence, specifically decreasing by 5.

Step 4: Calculate the Next Term

To find the next term in the sequence, subtract the common difference of 5 from the last term given, which is 47. So, the next term is: 47 - 5 = 42.

Step 5: Finalize the Solution

We've calculated the next term using the identified pattern. Confirm that it follows the pattern by checking backward from 42: 47, 42. This verifies the pattern is consistent.

Key Concepts

Patterns in Sequences

Understanding patterns in sequences is key in mathematics, especially when dealing with arithmetic sequences. A pattern in a sequence refers to the rule that defines how each term in the sequence is related to its predecessor. In the given problem, the sequence is 67, 62, 57, 52, 47. To identify a pattern, we need to find a common characteristic or rule that applies to all the terms. By observing each pair of terms in the sequence, we notice a consistent decrease. Recognizing these consistent changes helps us see the underlying pattern.

• Remember, each mathematical sequence often follows a specific type of pattern, which can be arithmetic, geometric, or something more complex.
• In arithmetic sequences, like the one in our problem, a constant difference exists between each term.

Understanding patterns is not just about identifying the numbers but also figuring out the rule that governs their appearance. This foundational step is crucial for predicting future terms in the sequence.

Consecutive Terms

Consecutive terms in a sequence are the numbers that follow one after another. Understanding how these terms are generated is vital to working with sequences effectively. In our sequence, we see the numbers are:

• 67 and 62;
• 62 and 57;
• 57 and 52;
• 52 and 47.

Each of these pairs represents consecutive terms. The consistency of the sequence is reflected in how these terms relate directly to each other, especially noticeable when looking at the arithmetic sequence. By focusing on consecutive terms, we can grow accustomed to how sequences evolve and immediately spot irregularities or confirmancies.

Subtraction in Sequences

Subtraction plays a crucial role in identifying arithmetic sequences. When we look at consecutive terms, we often compute the difference between them to understand the sequence's pattern. In the sequence provided, this difference is uniformly 5, as calculated through simple subtraction:

• 67 - 62
• 62 - 57
• 57 - 52
• 52 - 47

Each instance gives us a result of 5. Recognizing subtraction as a tool for pattern identification is essential, as it allows us to decide whether a sequence is arithmetic. Subtracting consecutive terms is almost like peeling back the layers to reveal the core structure of the sequence.

Mathematical Reasoning

Mathematical reasoning is the ability to think logically and systematically about numbers and patterns. It involves determining patterns, applying consistent operations, and predicting future terms. Within the context of our sequence, mathematical reasoning helps us not only calculate differences but also draw conclusions about the nature of the sequence. When faced with the sequence 67, 62, 57, 52, 47, we used logical deduction to:

• Identify that the pattern involves subtraction by 5.
• Recognize the sequence as arithmetic.
• Predict the next term based on this consistent rule.

Employing mathematical reasoning involves more than calculation; it requires a critical assessment of how and why. It allows problem-solving in a methodical way, enabling predictions and confirmations through step-by-step analysis.