Explanation:
To find the sum of the sequence \(1, 10, 100, 1000, \ldots, 1,000,000,000\), we note that it is a geometric sequence where:

- The first term \(a = 1\)
- The common ratio \(r = 10\)
- The last term \(l = 1,000,000,000\)

First, determine the number of terms in the sequence.

The general term of the geometric sequence can be written as:

\[ a_n = a \cdot r^{(n - 1)} \]

Set \(a_n = 1,000,000,000\):

\[ 1,000,000,000 = 1 \cdot 10^{(n - 1)} \]

\[ 10^{(n - 1)} = 1,000,000,000 \]

Since \(1,000,000,000 = 10^9\), we have:

\[ n - 1 = 9 \]

\[ n = 10 \]

So there are 10 terms in the sequence.

The sum \(S_n\) of the first \(n\) terms of a geometric sequence is given by:

\[ S_n = a \frac{r^n - 1}{r - 1} \]

Substituting the known values:

- \(a = 1\)
- \(r = 10\)
- \(n = 10\)

\[ S_{10} = 1 \frac{10^{10} - 1}{10 - 1} \]

\[ S_{10} = \frac{10^{10} - 1}{9} \]

Calculate \(10^{10} - 1\):

\[ 10^{10} = 10,000,000,000 \]

\[ 10^{10} - 1 = 9,999,999,999 \]

Now, divide by 9:

\[ \frac{9,999,999,999}{9} = 1,111,111,111 \]

Thus, the sum of the sequence is:

A. 11,111,111,111