To find the sum of a finite geometric series, we utilize the formula:
[tex]\[ S_n = a \frac{1 - r^n}{1 - r} \][/tex]
where:
- [tex]\(a\)[/tex] is the first term of the series,
- [tex]\(r\)[/tex] is the common ratio,
- [tex]\(n\)[/tex] is the number of terms.
Given the geometric series:
[tex]\[ \sum_{k=1}^9 4\left(-\frac{1}{3}\right)^k \][/tex]
we can identify the parameters as follows:
- The first term [tex]\(a\)[/tex] is 4.
- The common ratio [tex]\(r\)[/tex] is [tex]\(-\frac{1}{3}\)[/tex].
- The number of terms [tex]\(n\)[/tex] is 9.
Let's review and correctly apply the formula to the series sum:
The correct formula for the sum of the first [tex]\(n\)[/tex] terms in this case should be:
[tex]\[ S_9 = 4 \frac{1 - \left(-\frac{1}{3}\right)^9}{1 - \left(-\frac{1}{3}\right)} \][/tex]
Next, let’s choose the correct option based on this correct formula. We must identify any equivalent forms of the given formula:
1. [tex]\(\frac{-\frac{4}{3}\left(1-\left(-\frac{1}{3}\right)^8\right)}{1-\left(-\frac{1}{3}\right)}\)[/tex]
2. [tex]\(\frac{-\frac{4}{3}\left(1-\left(-\frac{1}{3}\right)^{10}\right)}{11}\)[/tex]
Analyzing the options:
- Option 1: [tex]\(\frac{-\frac{4}{3}\left(1-\left(-\frac{1}{3}\right)^8\right)}{1-\left(-\frac{1}{3}\right)}\)[/tex] is not correct because it misapplies the exponent [tex]\(8\)[/tex] instead of [tex]\(9\)[/tex], and the entire numerator structure is incorrect.
- Option 2: [tex]\(\frac{-\frac{4}{3}\left(1-\left(-\frac{1}{3}\right)^{10}\right)}{11}\)[/tex] is not correct because it adds an additional term in the exponent, wrong numerator, and incorrect denominator.
Given that neither option strictly matches the formula, one can deduce that either the options are incorrect, or there is a mistake in provided choices. Evaluating against correct formula:
[tex]\[ S_9 = 4 \frac{1 - \left(-\frac{1}{3}\right)^9}{1 - \left(-\frac{1}{3}\right)} = \frac{4 \left(1 - (-\frac{1}{3})^9\right)}{1 - (-\frac{1}{3})} = \frac{4 \left(1 - (-\frac{1}{3})^9\right)}{1 + \frac{1}{3}} = \frac{4 \left(1 - (-\frac{1}{3})^9\right)}{\frac{4}{3}} = 3 \left(1 - (-\frac{1}{3})^9\right) = 3.0001524157902764\][/tex]
So the detailed result aligns with the stepwise evaluated formula structure [tex]\(a = 4, r = - 1/3, n=9\)[/tex]:
Therefore, none provided are accurately tied to solved series sum as matching structure correct final validated formula and result.
