Find the common difference and the next three terms of each arithmetic sequence. 5. \( \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \ldots \)

To find the common difference and the next three terms of the arithmetic sequence, we can follow these steps: 1. Identify the first term of the sequence. 2. Find the common difference by subtracting the first term from the second term. 3. Use the common difference to find the next terms in the sequence. Step 1: Identify the first term. The first term of the sequence is \( \frac{1}{4} \). Step 2: Find the common difference. The second term is \( \frac{1}{2} \), and the first term is \( \frac{1}{4} \). To find the common difference, we subtract the first term from the second term: \[ \text{Common difference} = \frac{1}{2} - \frac{1}{4} \] To subtract these fractions, we need a common denominator, which is 4 in this case: \[ \text{Common difference} = \frac{2}{4} - \frac{1}{4} = \frac{1}{4} \] So, the common difference is \( \frac{1}{4} \). Step 3: Use the common difference to find the next three terms. To find the next term, we add the common difference to the last known term. We'll do this three times to find the next three terms. The last known term is \( \frac{5}{4} \). Next term: \[ \frac{5}{4} + \frac{1}{4} = \frac{6}{4} = \frac{3}{2} \] Second next term: \[ \frac{3}{2} + \frac{1}{4} = \frac{6}{4} + \frac{1}{4} = \frac{7}{4} \] Third next term: \[ \frac{7}{4} + \frac{1}{4} = \frac{8}{4} = 2 \] So, the common difference is \( \frac{1}{4} \), and the next three terms of the sequence are \( \frac{3}{2} \), \( \frac{7}{4} \), and 2.