Write the set `{(1)/(2),(2)/(5),(3)/(10),(4)/(17),(5)/(26),(6)/(37),(7)/(50),(8)/(65)}` in the Set - builder Form .

Answer

To express the given set \(\{ \frac{1}{2}, \frac{2}{5}, \frac{3}{10}, \frac{4}{17}, \frac{5}{26}, \frac{6}{37}, \frac{7}{50}, \frac{8}{65} \}\) in set-builder form, we need to identify a general rule that describes the elements of the set.

### Step-by-Step Solution:

1. **Identify the Numerators and Denominators**:
- The numerators of the fractions are the integers from 1 to 8.
- The denominators can be observed as \(2, 5, 10, 17, 26, 37, 50, 65\).

2. **Analyze the Denominators**:
- The denominators can be expressed as \(n^2 + 1\), where \(n\) is the numerator.
- For example:
- For \(n = 1\): \(1^2 + 1 = 2\)
- For \(n = 2\): \(2^2 + 1 = 5\)
- For \(n = 3\): \(3^2 + 1 = 10\)
- For \(n = 4\): \(4^2 + 1 = 17\)
- For \(n = 5\): \(5^2 + 1 = 26\)
- For \(n = 6\): \(6^2 + 1 = 37\)
- For \(n = 7\): \(7^2 + 1 = 50\)
- For \(n = 8\): \(8^2 + 1 = 65\)

3. **Formulate the Set-Builder Notation**:
- We can denote the set as \(A\) such that:
\[ A = \{ x \mid x = \frac{k}{k^2 + 1}, \, k \in \{1, 2, 3, 4, 5, 6, 7, 8\} \} \]

### Final Set-Builder Form:
Thus, the set in set-builder form is:
\[ A = \{ x \mid x = \frac{k}{k^2 + 1}, \, k = 1, 2, 3, 4, 5, 6, 7, 8 \} \]