Find the common difference and 99th term of the arithmetic progression :
`7 3/4, 9 1/2, 11 1/4,……….`

Answer

To solve the problem of finding the common difference and the 99th term of the arithmetic progression (AP) given as \(7 \frac{3}{4}, 9 \frac{1}{2}, 11 \frac{1}{4}, \ldots\), we will follow these steps:

### Step 1: Convert Mixed Fractions to Improper Fractions
First, we convert the mixed fractions into improper fractions for easier calculations.
- \(7 \frac{3}{4} = \frac{4 \times 7 + 3}{4} = \frac{28 + 3}{4} = \frac{31}{4}\)
- \(9 \frac{1}{2} = \frac{2 \times 9 + 1}{2} = \frac{18 + 1}{2} = \frac{19}{2} = \frac{38}{4}\) (to have a common denominator)
- \(11 \frac{1}{4} = \frac{4 \times 11 + 1}{4} = \frac{44 + 1}{4} = \frac{45}{4}\)

### Step 2: Identify the First Term and the Second Term
Now we identify the first term \(a\) and the second term \(a_2\):
- First term \(a = \frac{31}{4}\)
- Second term \(a_2 = \frac{38}{4}\)

### Step 3: Calculate the Common Difference
The common difference \(d\) of an arithmetic progression is found by subtracting the first term from the second term:
\[ d = a_2 - a = \frac{38}{4} - \frac{31}{4} = \frac{38 - 31}{4} = \frac{7}{4} \]

### Step 4: Convert the Common Difference to Mixed Fraction
To convert \(\frac{7}{4}\) into a mixed fraction:
\[ \frac{7}{4} = 1 \frac{3}{4} \]
Thus, the common difference \(d = 1 \frac{3}{4}\).

### Step 5: Find the 99th Term
The formula for the \(n\)th term of an arithmetic progression is given by:
\[ a_n = a + (n - 1) \cdot d \]
To find the 99th term \(a_{99}\):
\[ a_{99} = a + (99 - 1) \cdot d = a + 98 \cdot d \]
Substituting the values:
\[ a_{99} = \frac{31}{4} + 98 \cdot \frac{7}{4} \]
Calculating \(98 \cdot \frac{7}{4}\):
\[ 98 \cdot \frac{7}{4} = \frac{686}{4} \]
Now, adding:
\[ a_{99} = \frac{31}{4} + \frac{686}{4} = \frac{31 + 686}{4} = \frac{717}{4} \]

### Step 6: Convert the 99th Term to Mixed Fraction
To convert \(\frac{717}{4}\) into a mixed fraction:
\[ \frac{717}{4} = 179 \frac{1}{4} \]

### Final Answer
- The common difference \(d = 1 \frac{3}{4}\)
- The 99th term \(a_{99} = 179 \frac{1}{4}\)