Find the common difference of the A.P. `13, 15 1/2, 18, 20 1/2,………… . `

Answer

To find the common difference of the arithmetic progression (A.P.) given as \( 13, 15 \frac{1}{2}, 18, 20 \frac{1}{2}, \ldots \), 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 \( a = 13 \)
- Second term \( a_2 = 15 \frac{1}{2} = 15.5 \)
- Third term \( a_3 = 18 \)
- Fourth term \( a_4 = 20 \frac{1}{2} = 20.5 \)

### Step 2: Use the formula for the common difference
The common difference \( d \) in an A.P. can be calculated using the formula:
\[ d = a_2 - a_1 \]
where \( a_1 \) is the first term and \( a_2 \) is the second term.

### Step 3: Calculate the common difference
Now substituting the values:
\[ d = 15.5 - 13 \]
Calculating this gives:
\[ d = 15.5 - 13 = 2.5 \]

### Step 4: Verify using the next term
To ensure consistency, we can also check the difference between the third and second terms:
\[ d = a_3 - a_2 = 18 - 15.5 \]
Calculating this gives:
\[ d = 18 - 15.5 = 2.5 \]

### Step 5: Check the difference between the fourth and third terms
We can also check the difference between the fourth and third terms:
\[ d = a_4 - a_3 = 20.5 - 18 \]
Calculating this gives:
\[ d = 20.5 - 18 = 2.5 \]

### Conclusion
Thus, the common difference \( d \) of the A.P. is:
\[ \boxed{2.5} \]