Problem 34 - Thinking Mathematically

Question

In Exercises 9-38, identify a pattern in each list of numbers. Then use this pattern to find the next number. (More than one pattern might exist, so it is possible that there is more than one correct answer.) \(2,6,5,15,14,42,41,123\),_____

Step-by-Step Solution

Answer

The next number in the sequence is 369.

Step 1: Observe the pattern

The pattern begins with the number \(2\). Each subsequent number results from either multiplying the previous number by \(3\) or subtracting \(1\) from it. More clearly, the series follows the pattern \(ax3, -1, ax3, -1\), and so forth.

Step 2: Continue the pattern

To find the next number in the series, follow the pattern identified in Step 1. After \(2, 6, 5, 15, 14, 42, 41\), the next number is \(123\). Following the pattern, the next operation should be multiplication by \(3\). Doing this gives \(123 \times 3 = 369\).

Key Concepts

Understanding Number Sequences

Number sequences are essentially lists of numbers that follow a specific order or rule. They are crucial components in the field of mathematics, as they help in developing logical thinking and analytical skills. In the sequence provided: \(2, 6, 5, 15, 14, 42, 41, 123\), the task is to identify the underlying pattern that generates each subsequent number. Once the pattern is recognized, this enables us to predict future numbers in the sequence. When working with number sequences, it is important to look for consistent operations or changes applied from one number to the next.

Mastering Pattern Recognition

Pattern recognition involves identifying the rules or relationships among numbers, that repeat within a sequence. In the given exercise, the numbers alternate between operations of multiplying by 3 and subtracting 1.

• The first operation takes a number, multiplies it by 3, and results in the next number.
• The following operation subtracts 1 to reach the subsequent number.

For example, starting with 2: - Multiply by 3 to get \(2 \times 3 = 6\)- Subtract 1 to get \(6 - 1 = 5\)This continuous rule application allows us to predict that after 123, multiplying by 3 gives us the next number, 369.

Effective Problem Solving Strategies

Problem solving is a key skill that helps tackle mathematical challenges strategically. When asked to find the next number in a sequence, consider the following steps:

• Review the list of numbers to check for visible operations or repetitive patterns.
• Test small operations like addition, subtraction, multiplication, or division.
• Apply recognized patterns uniformly to confirm the rule and extend the sequence.

In the example, alternating multiplication and subtraction was identified as the rule. Thus, continuing this logical pathway yields the next number accurately when applied correctly.

Enhancing Mathematical Reasoning

Mathematical reasoning involves critically thinking about numbers and their relationships. It is an essential component to successfully identify and apply number patterns. Through mathematical reasoning, we systematically apply rule-based logic to understand sequences. This skill not only aids in finding correct sequence numbers but also sharpens cognitive abilities. To apply reasoning effectively:

• Approach each number sequence with an open mind for possible patterns.
• Evaluate each transformation critically to determine if it fits the identified rule.
• Always verify your result by checking if the calculated number fits logically into the sequence.

By utilizing these reasoning skills, students can solve problems more confidently and with greater accuracy.