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Chapter 1: Problem 34

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.) \(2,6,5,15,14,42,41,123\)

Short Answer

Expert verified
Following the identified pattern, the next number in the sequence should be 122.

Step by step solution

01

Pattern Observation

From observing the sequence, there might be two operations occurring alternatively. One is multiplication and the second seems to be subtraction. Let's break down the list into parts to further validate this hypothesis:\nStep 1: \(2 \times 3 = 6\)\nStep 2: \(6 - 1 = 5\)\nStep 3: \(5 \times 3 = 15\)\nStep 4: \(15 - 1 = 14\)\nStep 5: \(14 \times 3 = 42\)\nStep 6: \(42 - 1 = 41\)\nStep 7: \(41 \times 3 = 123\)\nAs per this pattern and based on the operations above, the sequence alternates between multiplying the current number by 3 and subtracting 1 from the result.
02

Implementing Pattern

Following this established pattern, for the next number in the sequence we need to subtract 1. Hence the next number will be \(123 - 1 = 122\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pattern Recognition
Recognizing patterns in a sequence of numbers is like finding a hidden code. You begin by carefully examining the numbers and try to understand how one number transitions into the next. In the given sequence
  • 2, 6, 5, 15, 14, 42, 41, 123
there's an underlying pattern that isn't immediately obvious. The first step is to check if there's a consistent change or operation when you move from one number to the next.
To solve this problem, look for common mathematical operations such as addition, subtraction, multiplication, or division.
Approach the sequence with curiosity, examining each transition to see if it involves one or multiple operations alternating throughout the list.
Alternating Operations
Alternating operations mean that different operations take turns being applied to create the next number in the sequence. In this example, the two alternating operations are multiplication and subtraction.
Notice how the sequence switches between multiplying and subtracting:
  • Multiply by 3 to transition from 2 to 6.
  • Subtract 1 to go from 6 to 5.
  • Multiply by 3 to transition from 5 to 15.
  • Subtract 1 to go from 15 to 14.
  • Multiply by 3 to transition from 14 to 42.
  • Subtract 1 to go from 42 to 41.
The key here is tracing the steps and seeing how these operations alternate, establishing a predictable pattern that can be applied to any part of the sequence. For students, recognizing alternating operations helps in understanding complex sequences not defined by a single operation.
Multiplication
Multiplication is one of the core operations in this number sequence. Multiplying a number by another changes its size, scaling it either up or down. In our pattern:
  • The process starts with multiplying each number by 3. E.g., 2 becomes 6 (since 2 \(\times\) 3 is 6).
  • Continuing this multiplication, 5 becomes 15, and later on, 14 becomes 42.
Understanding multiplication means seeing how it is used to grow or scale up each number at particular points in the sequence. By pinpointing which numbers to multiply, we maintain the sequence's structure. Ensure you grasp why multiplication at these steps maintains the alternating pattern.
Subtraction
Subtraction is the operation of removing a number from another. In this sequence, subtraction alternates with multiplication to create a unique pattern. Each subtraction step is straightforward:
  • After multiplying, the following number subtracts 1. E.g., 6 becomes 5 with 6 - 1 = 5.
  • Likewise, 15 becomes 14, 42 becomes 41, etc.
The subtraction is simple but crucial as it balances the multiplication operation. It gently reduces the number, preparing it for the next sequence multiplication. Understanding these alternating steps in subtraction helps us fill in the missing parts of the sequence. It teaches how to apply basic math operations in practical situations. By fully grasping subtraction's role, students can predict subsequent values in an alternating pattern sequence.

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