To find the sum of the first 10 terms of the given geometric series, we can use the formula for the sum of the first [tex]\(n\)[/tex] terms of a geometric series, which is:
[tex]\[
S_n = \frac{a_1\left(1-r^n\right)}{1-r}
\][/tex]
Where:
- [tex]\(S_n\)[/tex] is the sum of the first [tex]\(n\)[/tex] terms,
- [tex]\(a_1\)[/tex] is the first term of the series,
- [tex]\(r\)[/tex] is the common ratio, and
- [tex]\(n\)[/tex] is the number of terms.
Given the series: [tex]\(6,144 + 3,072 + 1,536 + 768 + \cdots\)[/tex], we can identify the following:
- The first term [tex]\(a_1 = 6,144\)[/tex],
- The common ratio [tex]\(r\)[/tex] can be determined by dividing the second term by the first term, [tex]\(r = \frac{3,072}{6,144} = 0.5\)[/tex],
- The number of terms [tex]\(n = 10\)[/tex].
Substituting these values into the formula:
[tex]\[
S_{10} = \frac{6,144 \left(1 - (0.5)^{10}\right)}{1 - 0.5}
\][/tex]
Now, we can simplify the calculation step by step:
1. Calculate the denominator:
[tex]\[
1 - 0.5 = 0.5
\][/tex]
2. Calculate the [tex]\(n\)[/tex]th power of the common ratio:
[tex]\[
(0.5)^{10} = 0.0009765625
\][/tex]
3. Subtract this value from 1:
[tex]\[
1 - 0.0009765625 = 0.9990234375
\][/tex]
4. Multiply this result by the first term:
[tex]\[
6,144 \times 0.9990234375 = 6,138
\][/tex]
5. Finally, divide by the denominator:
[tex]\[
\frac{6,138}{0.5} = 12,276
\][/tex]
Thus, the sum of the first 10 terms of this geometric series is:
[tex]\[
\boxed{12,276}
\][/tex]
So, the correct answer is:
B. 12,276
