Find the common difference of the A.P. `5, 6 1/2, 8, 9 1/2`,…………. .

Answer

To find the common difference of the arithmetic progression (A.P.) given as \(5, 6 \frac{1}{2}, 8, 9 \frac{1}{2}, \ldots\), we can follow these steps:

### Step 1: Identify the terms of the A.P.
The first few terms of the A.P. are:
- First term (\(a_1\)) = \(5\)
- Second term (\(a_2\)) = \(6 \frac{1}{2} = \frac{13}{2}\)
- Third term (\(a_3\)) = \(8\)
- Fourth term (\(a_4\)) = \(9 \frac{1}{2} = \frac{19}{2}\)

### Step 2: Calculate the common difference (\(d\))
The common difference (\(d\)) in an A.P. can be calculated by subtracting the first term from the second term or the second term from the third term, and so on.

#### Calculation:
1. **Using the first two terms:**
\[ d = a_2 - a_1 = \frac{13}{2} - 5 \]
To perform the subtraction, convert \(5\) into a fraction:
\[ 5 = \frac{10}{2} \]
Now substitute:
\[ d = \frac{13}{2} - \frac{10}{2} = \frac{13 - 10}{2} = \frac{3}{2} \]

2. **Using the second and third terms:**
\[ d = a_3 - a_2 = 8 - \frac{13}{2} \]
Convert \(8\) into a fraction:
\[ 8 = \frac{16}{2} \]
Now substitute:
\[ d = \frac{16}{2} - \frac{13}{2} = \frac{16 - 13}{2} = \frac{3}{2} \]

3. **Using the third and fourth terms:**
\[ d = a_4 - a_3 = \frac{19}{2} - 8 \]
Convert \(8\) into a fraction:
\[ 8 = \frac{16}{2} \]
Now substitute:
\[ d = \frac{19}{2} - \frac{16}{2} = \frac{19 - 16}{2} = \frac{3}{2} \]

### Conclusion:
The common difference \(d\) of the A.P. is \(\frac{3}{2}\).

### Final Answer:
The common difference of the A.P. is \(\frac{3}{2}\).

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