Problem 27 - 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.) \(3,6,11,18,27,38\),_____

Step-by-Step Solution

Answer

The next number in the sequence is \(51\).

Step 1: Identify the pattern in the numbers

To find the pattern, subtract each pair of continuous numbers. The difference between successive numbers is as follows: \(6 - 3 = 3\), \(11 - 6 = 5\), \(18 - 11 = 7\), \(27 - 18 = 9\), \(38 - 27 = 11\). So, the differences are successively increasing by \(2\).

Step 2: Apply the pattern to find the next number

Since the differences between successive numbers are increasing by \(2\), the difference between the next number and \(38\) should be \(11 + 2 = 13\). Therefore, the next number should be \(38 + 13 = 51\).

Key Concepts

Number Sequences

Number sequences are a fundamental part of mathematics and can often seem like puzzles waiting to be solved. They are essentially lists of numbers that follow a certain rule or pattern. Understanding number sequences is crucial because they are the backbone of more complex mathematical concepts and appear in various forms across different areas of mathematics and science.

A sequence might be as simple as counting numbers – 1, 2, 3, 4 ... or more complex with varying increments, like the sequence provided in the exercise: 3, 6, 11, 18, 27, 38. Sequences can have practical applications too, such as calculating interest, predicting population growth, or even in computer algorithms. When faced with a sequence, the goal is to decode the underlying rule and predict subsequent numbers.

Pattern Recognition

Pattern recognition in mathematics involves identifying regularities and systematic relationships within numbers, shapes, or sets of data. It's a skill that taps into analytical thinking, allowing us to predict and extrapolate information.

In the given exercise, pattern recognition is key to solving the sequence. By investigating the differences between successive numbers, we can detect that each time we're adding successively larger even numbers. This insight allows us to extend the sequence reliably. Pattern recognition isn't just mathematics – it's a fundamental cognitive skill we use daily, from recognizing faces to interpreting music.

Arithmetic Sequences

An arithmetic sequence is a number sequence in which the difference (called the 'common difference') between consecutive terms is constant. This sequence typically has a form where each term is the previous one plus a fixed number. For example, the sequence 2, 4, 6, 8, ... increases by 2 each time, making it a simple arithmetic sequence.

The sequence in the exercise, however, is not a simple arithmetic sequence since the difference between terms continually increases by 2, not repeating the same incremental value. Despite this, understanding arithmetic sequences aids in solving and grasping more complex patterns, revealing how essential these foundational concepts are in breaking down and solving advanced mathematical problems.