To evaluate the given geometric series [tex]\(\sum_{k=1}^8 5\left(\frac{2}{3}\right)^{k-1}\)[/tex], we need to recognize the form of this series. A geometric series has the form:
[tex]\[
\sum_{k=0}^{n-1} ar^k = a \left(\frac{1 - r^n}{1 - r}\right)
\][/tex]
In this problem:
- The first term, [tex]\( a = 5 \)[/tex]
- The common ratio, [tex]\( r = \frac{2}{3} \)[/tex]
- The series runs from [tex]\( k = 1 \)[/tex] to [tex]\( k = 8\)[/tex], which corresponds to [tex]\( k - 1 = 0 \)[/tex] to [tex]\( 7 \)[/tex].
Thus, the correct formula for the sum of the first 8 terms of the series is:
[tex]\[
5 \left(\frac{1 - \left(\frac{2}{3}\right)^8}{1 - \frac{2}{3}}\right)
\][/tex]
We evaluate the other given options as well:
1. [tex]\( 5 \left(\frac{1 - \left(\frac{2}{3}\right)^8}{1 - \frac{2}{3}}\right) \)[/tex]:
Evaluating the formula:
[tex]\[
5 \left(\frac{1 - \left(\frac{2}{3}\right)^8}{1 - \frac{2}{3}}\right) = 14.414723365340649
\][/tex]
2. [tex]\( 5 \left(\frac{1 - \left(\frac{2}{3}\right)^7}{1 - \frac{2}{3}}\right) \)[/tex]:
This formula represents the sum of the first 7 terms of the series:
[tex]\[
5 \left(\frac{1 - \left(\frac{2}{3}\right)^7}{1 - \frac{2}{3}}\right) = 14.122085048010973
\][/tex]
3. [tex]\( 5 \left(\frac{1 - \left(\frac{2}{3}\right)^9}{1 - \frac{2}{9}}\right) \)[/tex]:
This does not properly represent the sum of a geometric series with the given common ratio and also is computed as:
[tex]\[
5 \left(\frac{1 - \left(\frac{2}{3}\right)^9}{1 - \frac{2}{9}}\right) = 6.261349532954471
\][/tex]
Upon comparing the results:
- The first formula [tex]\( 5 \left(\frac{1 - \left(\frac{2}{3}\right)^8}{1 - \frac{2}{3}}\right) \)[/tex] accurately represents the sum of the given series [tex]\(\sum_{k=1}^8 5\left(\frac{2}{3}\right)^{k-1}\)[/tex].
Therefore, the formula that can be used to evaluate the series [tex]\(\sum_{k=1}^8 5\left(\frac{2}{3}\right)^{k-1}\)[/tex] is:
[tex]\[
\boxed{5\left(\frac{1 - \left(\frac{2}{3}\right)^8}{1 - \frac{2}{3}}\right)}
\][/tex]
