What is the sum of the sequence: 1, 10, 100, 1000, ... 1, 000, 000, 000?
A. 11,111,111,11 B. 11,111,111,90 C. 99,999,999,11 D. 99,999,999,99 Correct Answer: AExplanationTo 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 |