Which of the following are AP's ? If they form an A. P. , find the common difference d and write three more terms.
`2, (5)/(2), 3, (7)/(2)…`

Answer

To determine if the given sequence \(2, \frac{5}{2}, 3, \frac{7}{2}, \ldots\) forms an arithmetic progression (AP), we need to check if the common difference \(d\) between consecutive terms is constant.

### Step-by-Step Solution:

1. **Identify the terms**:
- Let \(a_1 = 2\)
- Let \(a_2 = \frac{5}{2}\)
- Let \(a_3 = 3\)
- Let \(a_4 = \frac{7}{2}\)

2. **Calculate the common difference \(d\)**:
- The common difference \(d\) can be calculated as:
\[ d = a_2 - a_1 = \frac{5}{2} - 2 \]
- To perform the subtraction, convert \(2\) into a fraction with a denominator of \(2\):
\[ 2 = \frac{4}{2} \]
- Now, substitute:
\[ d = \frac{5}{2} - \frac{4}{2} = \frac{1}{2} \]

3. **Check the next common difference**:
- Calculate \(d\) again using \(a_3\) and \(a_2\):
\[ d = a_3 - a_2 = 3 - \frac{5}{2} \]
- Convert \(3\) into a fraction:
\[ 3 = \frac{6}{2} \]
- Now, substitute:
\[ d = \frac{6}{2} - \frac{5}{2} = \frac{1}{2} \]

4. **Check the next common difference**:
- Calculate \(d\) using \(a_4\) and \(a_3\):
\[ d = a_4 - a_3 = \frac{7}{2} - 3 \]
- Convert \(3\) into a fraction:
\[ 3 = \frac{6}{2} \]
- Now, substitute:
\[ d = \frac{7}{2} - \frac{6}{2} = \frac{1}{2} \]

5. **Conclusion about AP**:
- Since the common difference \(d\) is the same for all pairs of terms, the sequence forms an AP with \(d = \frac{1}{2}\).

6. **Finding the next three terms**:
- The \(n\)-th term of an AP can be calculated using the formula:
\[ a_n = a + (n-1)d \]
- For \(a_5\):
\[ a_5 = a + 4d = 2 + 4 \cdot \frac{1}{2} = 2 + 2 = 4 \]
- For \(a_6\):
\[ a_6 = a + 5d = 2 + 5 \cdot \frac{1}{2} = 2 + \frac{5}{2} = \frac{4}{2} + \frac{5}{2} = \frac{9}{2} \]
- For \(a_7\):
\[ a_7 = a + 6d = 2 + 6 \cdot \frac{1}{2} = 2 + 3 = 5 \]

7. **Final answer**:
- The next three terms are \(4\), \(\frac{9}{2}\), and \(5\).