Finding the Next Fraction in the Series

The question asks us to find the next fraction in the given series:

\(\frac{2}{3}, \frac{4}{5}, \frac{6}{7}, \frac{8}{9}, \frac{10}{11}, \frac{12}{13}, \dots\)

To determine the next fraction, we need to identify the pattern in the sequence of fractions.

Analyzing the Pattern in the Series

Let's examine the numerators and denominators of the fractions separately.

Pattern in Numerators

The sequence of numerators is:

\(2, 4, 6, 8, 10, 12, \dots\)

Observe the difference between consecutive terms:

• \(4 - 2 = 2\)
• \(6 - 4 = 2\)
• \(8 - 6 = 2\)
• \(10 - 8 = 2\)
• \(12 - 10 = 2\)

The numerators form an arithmetic progression with a common difference of \(2\). To find the next numerator, we add \(2\) to the last numerator in the given series, which is \(12\).

\(\text{Next numerator} = 12 + 2 = 14\)

Pattern in Denominators

The sequence of denominators is:

\(3, 5, 7, 9, 11, 13, \dots\)

Observe the difference between consecutive terms:

• \(5 - 3 = 2\)
• \(7 - 5 = 2\)
• \(9 - 7 = 2\)
• \(11 - 9 = 2\)
• \(13 - 11 = 2\)

The denominators also form an arithmetic progression with a common difference of \(2\). To find the next denominator, we add \(2\) to the last denominator in the given series, which is \(13\).

\(\text{Next denominator} = 13 + 2 = 15\)

Determining the Next Fraction

The next fraction in the series will have the next numerator as its numerator and the next denominator as its denominator.

\(\text{Next fraction} = \frac{\text{Next numerator}}{\text{Next denominator}} = \frac{14}{15}\)

Comparing with Options

Let's compare the calculated next fraction with the given options:

The calculated next fraction, \(\frac{14}{15}\), matches Option 1.

Conclusion on Fraction Series Pattern

The series follows a clear pattern where both the numerator and the denominator increase by \(2\) for each successive term. The numerator starts at \(2\), and the denominator starts at \(3\).

Revision Table: Fraction Series Patterns

Additional Information: Types of Sequences

Sequences, whether of numbers or parts of fractions, can follow various patterns. Identifying the pattern is key to predicting future terms.

• Arithmetic Sequence: Each term after the first is obtained by adding a constant number (the common difference) to the previous term. Example: \(5, 10, 15, 20, \dots\) (common difference = 5).
• Geometric Sequence: Each term after the first is obtained by multiplying the previous term by a constant number (the common ratio). Example: \(2, 4, 8, 16, \dots\) (common ratio = 2).
• Fibonacci Sequence: Each term is the sum of the two preceding terms (starting with 0 and 1 or 1 and 1). Example: \(1, 1, 2, 3, 5, 8, \dots\).
• Other Patterns: Sequences can follow patterns based on squares, cubes, prime numbers, or combinations of operations.

In this specific fraction series question, both the numerator and denominator sequences were simple arithmetic progressions.