What is 0.9+0.09+0.009+ . . . Equal to ?

Options

A. 1
B. `1.01`
C. `1.001`
D. `1.1`

Answer

To solve the problem \(0.9 + 0.09 + 0.009 + \ldots\), we can recognize that this is an infinite geometric series. Let's break it down step by step.

### Step 1: Identify the series
The series can be expressed as:
\[ S = 0.9 + 0.09 + 0.009 + \ldots \]
This can be rewritten as:
\[ S = 9 \times 0.1 + 9 \times 0.01 + 9 \times 0.001 + \ldots \]

### Step 2: Factor out the common term
We can factor out the common term \(9\):
\[ S = 9 \left(0.1 + 0.01 + 0.001 + \ldots\right) \]

### Step 3: Identify the first term and common ratio
Now, let's analyze the series inside the parentheses:
\[ T = 0.1 + 0.01 + 0.001 + \ldots \]
Here, the first term \(a = 0.1\) and the common ratio \(r = \frac{0.01}{0.1} = \frac{1}{10}\).

### Step 4: Use the formula for the sum of an infinite geometric series
The sum \(T\) of an infinite geometric series can be calculated using the formula:
\[ T = \frac{a}{1 - r} \]
Substituting the values we have:
\[ T = \frac{0.1}{1 - \frac{1}{10}} = \frac{0.1}{\frac{9}{10}} = 0.1 \times \frac{10}{9} = \frac{1}{9} \]

### Step 5: Substitute back to find \(S\)
Now, substituting \(T\) back into the equation for \(S\):
\[ S = 9 \times T = 9 \times \frac{1}{9} = 1 \]

### Conclusion
Thus, the sum of the series \(0.9 + 0.09 + 0.009 + \ldots\) is equal to:
\[ \boxed{1} \]