To solve the series, we need to evaluate the sum:
[tex]\[
\sum_{k=1}^8 5\left(\frac{4}{3}\right)^{k-1}
\][/tex]
This is a geometric series with the first term [tex]\( a = 5 \)[/tex] and the common ratio [tex]\( r = \frac{4}{3} \)[/tex].
The sum of the first [tex]\( n \)[/tex] terms of a geometric series is given by the formula:
[tex]\[
S_n = a \frac{1 - r^n}{1 - r}
\][/tex]
Plugging in the given values:
[tex]\[
a = 5, \quad r = \frac{4}{3}, \quad n = 8
\][/tex]
Using these values in the formula, we get:
[tex]\[
S_8 = 5 \frac{1 - \left(\frac{4}{3}\right)^8}{1 - \frac{4}{3}}
\][/tex]
Now, evaluating the expression within the series sum involves more detailed computation. After performing the exact calculations, the result of the sum is:
[tex]\[
S_8 \approx 134.83
\][/tex]
Therefore, the correct answer is:
D. 134.83
