Problem 22 - 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.) \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}\),_____

Step-by-Step Solution

Answer

The next number in the sequence is \(\frac{1}{6}\).

Step 1: Observing The Pattern

By observing the list of numbers, one can see that it consists of fractions. Each fraction has 1 as the numerator and the denominator increases by 1 for each subsequent term. The first term is \(1 = \frac{1}{1}\), the second term is \(\frac{1}{2}\), the third term is \(\frac{1}{3}\), the fourth term is \(\frac{1}{4}\), and the fifth term is \(\frac{1}{5}\).

Step 2: Predicting the Next Term

If we follow this pattern, then the denominator of the next term should be the next natural number after 5. Therefore, the sixth term will be \(\frac{1}{6}\).

Key Concepts

Fractions in Sequences

Fractions are a fundamental part of mathematics, representing a division of a whole into parts. They have a numerator and a denominator separated by a slash. In the exercise, fractions are used in a specific sequence. This sequence starts with a whole number 1, expressed as the fraction \( \frac{1}{1} \). Each term in the sequence follows by decreasing the denominator while keeping the numerator constant at 1. By doing so, the value of the fraction decreases with each step, moving from

• \( \frac{1}{2} \) to \( \frac{1}{3} \),
• \( \frac{1}{3} \) to \( \frac{1}{4} \), and,
• \( \frac{1}{4} \) to \( \frac{1}{5} \).

This pattern highlights the concept of reducing fraction values by increasing the denominator, portraying how division results impact numerical values.
In sequences like this, each step signifies multiplying the whole by inverses of natural numbers, demonstrating fractional reductions.

Exploring Numerical Sequences

A numerical sequence is a set of numbers arranged in a specific order following a particular rule. Recognizing and understanding these rules is crucial for finding missing elements in the sequence. In our current example, the sequence is defined by fractions with descending numeric values, beginning with whole number 1. The given sequence, though expressed in fractions, follows a clear pattern: the denominators are consecutive natural numbers.
These sequences give insight into how numbers can behave in structured orders:

• The first number is \( 1 \) which can be seen as \( \frac{1}{1} \).
• It continues with \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5} \)

Each sequence's term is based on increasing the denominator by one while keeping the numerator constant at one, demonstrating a logical progression of natural numbers as denominators.

Pattern Recognition Skills

Identifying patterns involves analyzing a sequence of numbers to find rules or formulas governing the sequence. Pattern recognition helps in predicting the subsequent terms in the sequence. For the given sequence \( 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5} \), the pattern is clear – each term's numerator remains 1, but the denominator increases by one each time.
Developing pattern recognition skills requires:

• Observation – Carefully observing changes and relationships between numbers.
• Practice – Solving various sequences to become familiar with different patterns.
• Logic – Understanding the basic principles of arithmetic that might explain a sequence's rule.

By recognizing patterns, students can simplify and solve complex numerical sequences. In this exercise, the missing term is easily filled once the pattern is seen: the next natural number after 5 is 6, making the next term \( \frac{1}{6} \). Evaluating each sequence part teaches both structural and analytical thinking skills, essential for broader mathematical reasoning.