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What is the sum of the first 10 terms of this geometric series? Use [tex]$S_n=\frac{a_1\left(1-r^n\right)}{1-r}$[/tex].

[tex]6,144+3,072+1,536+768+\cdots[/tex]

A. 11,520
B. 12,276
C. 23,040
D. 24,550


Sagot :

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