If a = 0.1818181818… and b = 0.3030030003 … then (a+b) is :

Options

A. A)a rational no.
B. B)a perfect number
C. C)an irrational no.
D. D)both (b) and (c)

Answer

To solve the problem of finding \( a + b \) where \( a = 0.1818181818\ldots \) and \( b = 0.3030030003\ldots \), we can follow these steps:

### Step 1: Convert \( a \) into a Fraction
The number \( a \) can be expressed as a repeating decimal:
\[ a = 0.1818181818\ldots = 0.\overline{18} \]
To convert this repeating decimal into a fraction, we can use the formula for converting repeating decimals:
Let \( x = 0.\overline{18} \).
Then, multiplying both sides by 100 (since the repeating part has 2 digits):
\[ 100x = 18.1818181818\ldots \]
Now, subtract the first equation from this new equation:
\[ 100x - x = 18.1818181818\ldots - 0.1818181818\ldots \]
This simplifies to:
\[ 99x = 18 \]
Thus, we can solve for \( x \):
\[ x = \frac{18}{99} = \frac{2}{11} \]
So, we have:
\[ a = \frac{2}{11} \]

### Step 2: Analyze \( b \)
Next, we look at \( b = 0.3030030003\ldots \). This number has a pattern where the digits '3' and '0' repeat, but the number of zeros increases. It can be expressed as:
\[ b = 0.3 + 0.003 + 0.000003 + \ldots \]
This is an infinite series. The first term is \( 0.3 \) and the second term is \( 0.003 \), which can be expressed as:
\[ b = 0.3 + \frac{3}{1000} + \frac{3}{1000000} + \ldots \]
This series can be recognized as a geometric series where the first term \( a = 0.3 \) and the common ratio \( r = \frac{1}{1000} \).

### Step 3: Sum the Series for \( b \)
The sum \( S \) of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
Substituting the values:
\[ S = \frac{0.3}{1 - \frac{1}{1000}} = \frac{0.3}{\frac{999}{1000}} = 0.3 \cdot \frac{1000}{999} = \frac{300}{999} = \frac{100}{333} \]
Thus, we have:
\[ b = \frac{100}{333} \]

### Step 4: Add \( a \) and \( b \)
Now we can add \( a \) and \( b \):
\[ a + b = \frac{2}{11} + \frac{100}{333} \]
To add these fractions, we need a common denominator. The least common multiple of \( 11 \) and \( 333 \) is \( 333 \):
\[ \frac{2}{11} = \frac{2 \times 30.2727}{11 \times 30.2727} = \frac{60.5454}{333} \]
Now we can add:
\[ a + b = \frac{60.5454}{333} + \frac{100}{333} = \frac{160.5454}{333} \]

### Step 5: Determine the Nature of \( a + b \)
Since \( a \) is a rational number and \( b \) is a repeating decimal which can be expressed as a fraction, the sum \( a + b \) is also a rational number.

### Conclusion
Thus, \( a + b \) is a rational number.