Find the common difference of A.P `4(1/9), 4(2/9), 4(1/3)`…………

Answer

To find the common difference of the arithmetic progression (A.P) given by the terms \(4\frac{1}{9}, 4\frac{2}{9}, 4\frac{1}{3}\), we will follow these steps:

### Step 1: Identify the terms
The first term \(a_1\) is \(4\frac{1}{9}\) and the second term \(a_2\) is \(4\frac{2}{9}\).

### Step 2: Convert mixed fractions to improper fractions
We will convert both mixed fractions to improper fractions for easier calculation.

1. For \(4\frac{1}{9}\):
\[ 4\frac{1}{9} = 4 + \frac{1}{9} = \frac{4 \times 9 + 1}{9} = \frac{36 + 1}{9} = \frac{37}{9} \]

2. For \(4\frac{2}{9}\):
\[ 4\frac{2}{9} = 4 + \frac{2}{9} = \frac{4 \times 9 + 2}{9} = \frac{36 + 2}{9} = \frac{38}{9} \]

### Step 3: Calculate the common difference
Now, we can find the common difference \(d\) by subtracting the first term from the second term:
\[ d = a_2 - a_1 = \frac{38}{9} - \frac{37}{9} \]

### Step 4: Perform the subtraction
Since both fractions have the same denominator, we can subtract the numerators:
\[ d = \frac{38 - 37}{9} = \frac{1}{9} \]

### Conclusion
The common difference of the A.P is \(\frac{1}{9}\).

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