\( \frac { 2 } { 2 } , \frac { 4 } { 5 } , \frac { 6 } { 8 } , \frac { 8 } { 11 } \)

The sequence you provided consists of fractions: \[ \frac{2}{2}, \frac{4}{5}, \frac{6}{8}, \frac{8}{11} \] Let's analyze the numerators and denominators separately. **Numerators:** - The numerators are: 2, 4, 6, 8. - These form an arithmetic sequence where each term increases by 2. **Denominators:** - The denominators are: 2, 5, 8, 11. - These also form an arithmetic sequence where each term increases by 3. Now, we can express the \(n\)-th term of each sequence: 1. For the numerators: - The \(n\)-th term can be expressed as \(2n\) (where \(n = 1, 2, 3, 4\)). 2. For the denominators: - The \(n\)-th term can be expressed as \(3n - 1\) (where \(n = 1, 2, 3, 4\)). Thus, the \(n\)-th term of the sequence of fractions can be written as: \[ \frac{2n}{3n - 1} \] This formula generates the fractions you provided for \(n = 1, 2, 3, 4\): - For \(n = 1\): \(\frac{2 \cdot 1}{3 \cdot 1 - 1} = \frac{2}{2}\) - For \(n = 2\): \(\frac{2 \cdot 2}{3 \cdot 2 - 1} = \frac{4}{5}\) - For \(n = 3\): \(\frac{2 \cdot 3}{3 \cdot 3 - 1} = \frac{6}{8}\) - For \(n = 4\): \(\frac{2 \cdot 4}{3 \cdot 4 - 1} = \frac{8}{11}\) If you have any specific questions or need further analysis, feel free to ask!