Problem 74 - Algebra 1: Concepts and Skills

Question

List the next three numbers suggested by the sequence. (Skills Review pp. 781) $$ \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, ?, ?, ? $$

Step-by-Step Solution

Answer

The next three terms in the sequence are \(\frac{5}{6}\), \(\frac{6}{7}\), and \(\frac{7}{8}\)

Step 1: Identify the Pattern

Observe the given fractions and identify the pattern. Notice that the numerator is one less than the denominator for each given fraction.

Step 2: Write The General Term for the Sequence

Based on the observation, each term of the sequence can be expressed in the form of \(\frac{n}{n+1}\), where \(n\) is the term number starting from 1.

Step 3: Calculate the next three terms

Substitute values for \(n\) with 5, 6 and 7 which will give the 5th, 6th and 7th terms of sequence: \(\frac{5}{6}\), \(\frac{6}{7}\), and \(\frac{7}{8}\) are hence the next three numbers in this sequence.

Key Concepts

Understanding Fractions

A fraction represents a part of a whole or, more generally, any number of equal parts. It's shown as \[\frac{a}{b}\]where \(a\) is the numerator—the number of parts you have—and \(b\) is the denominator—the number of parts the whole is divided into.
Fractions are everywhere in mathematics, from simple calculations to complex theories. They are used to explain and solve problems where division is key. They form the basis for understanding rational numbers and operations with them.

In the context of arithmetic sequences and series, fractions assist in demonstrating repeating patterns and relationships. Recognizing patterns in fraction series can significantly enhance your comprehension of arithmetic sequences and help in finding subsequent terms.

Exploring Number Patterns

Number patterns are sequences of numbers that develop based on a specific rule. Identifying patterns in numbers can help make educated predictions, determine underlying rules, and distinguish relationships between numbers.
In arithmetic sequences like the example given, number patterns appear as a specific, predictable sequence.
In our fraction sequence, observe that:

• The numerator increases by 1 in each subsequent fraction.
• The denominator is always 1 more than the numerator.

This uniform increase indicates a pattern which is crucial when predicting or calculating the next terms in the series. Understanding these patterns turns complex-looking sequences into something much simpler to tackle.
Number patterns are a handy tool not just in solving problems, but also in developing a deeper understanding of mathematical concepts.

Diving into Sequences and Series

Sequences and series are foundational concepts in mathematics, used to identify and predict relationships among numbers. A sequence is essentially an ordered list defined by a specific rule.
An arithmetic sequence is one where each term after the first is derived by adding a constant to the previous term. For our example of fractions \[\left(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots \right)\]it's clear each term follows a pattern described by the general formula:\[\frac{n}{n+1}\]where \(n\) starts from 1 and increases sequentially.

Recognizing this, you can now find any term in the sequence just by substituting the appropriate value of \(n\). A series is the sum of the terms of a sequence. While our current focus is sequences, both concepts offer powerful tools in problem-solving and mathematical reasoning.