If N=1 + 11 + 101 + 1001 + ... + 1000000001, what is the value of N ?

Options

A. 1111111100
B. 1111111110
C. 1111111120
D. 1111111111

Answer

To find the value of \( N = 1 + 11 + 101 + 1001 + ... + 1000000001 \), we can observe the pattern in the numbers being added.

### Step-by-Step Solution:

1. **Identify the Pattern**:
The numbers can be expressed as:
- \( 1 = 1 \)
- \( 11 = 10 + 1 \)
- \( 101 = 100 + 1 \)
- \( 1001 = 1000 + 1 \)
- ...
- \( 1000000001 = 10^9 + 1 \)

Each term can be represented as \( 10^k + 1 \) where \( k \) ranges from \( 0 \) to \( 9 \).

2. **Count the Terms**:
There are 10 terms in total, from \( k = 0 \) to \( k = 9 \).

3. **Separate the Terms**:
We can separate the \( 1 \) from each term:
\[ N = (1 + 1 + 1 + ... + 1) + (10^0 + 10^1 + 10^2 + ... + 10^9) \]
The first part has \( 10 \) ones (since there are 10 terms):
\[ 1 + 1 + 1 + ... + 1 = 10 \]

4. **Sum the Powers of 10**:
The second part is the sum of powers of \( 10 \):
\[ 10^0 + 10^1 + 10^2 + ... + 10^9 \]
This is a geometric series with the first term \( a = 1 \) (which is \( 10^0 \)) and the common ratio \( r = 10 \). The formula for the sum of the first \( n \) terms of a geometric series is:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Here, \( n = 10 \):
\[ S_{10} = 1 \cdot \frac{10^{10} - 1}{10 - 1} = \frac{10^{10} - 1}{9} \]

5. **Combine the Results**:
Now we can combine both parts:
\[ N = 10 + \frac{10^{10} - 1}{9} \]

6. **Simplify**:
To combine \( 10 \) with the fraction, we can express \( 10 \) as \( \frac{90}{9} \):
\[ N = \frac{90}{9} + \frac{10^{10} - 1}{9} = \frac{90 + 10^{10} - 1}{9} = \frac{10^{10} + 89}{9} \]

### Final Answer:
Thus, the value of \( N \) is:
\[ N = \frac{10^{10} + 89}{9} \]