To determine the sum of the given series [tex]\( \sum_{k=1}^8 5\left(\frac{4}{3}\right)^{(k-1)} \)[/tex], let's break down the problem step by step.
Firstly, identify the type of series presented. This is a geometric series where:
- The first term [tex]\( a \)[/tex] is 5.
- The common ratio [tex]\( r \)[/tex] is [tex]\( \frac{4}{3} \)[/tex].
The formula to find the sum [tex]\( S_n \)[/tex] of the first [tex]\( n \)[/tex] terms of a geometric series is:
[tex]\[
S_n = \frac{a(1 - r^n)}{1 - r}
\][/tex]
Given that:
- [tex]\( a = 5 \)[/tex]
- [tex]\( r = \frac{4}{3} \)[/tex]
- [tex]\( n = 8 \)[/tex]
We substitute these values into the formula:
[tex]\[
S_8 = \frac{5(1 - \left(\frac{4}{3}\right)^8)}{1 - \frac{4}{3}}
\][/tex]
The denominator [tex]\( 1 - \frac{4}{3} \)[/tex] simplifies to [tex]\( -\frac{1}{3} \)[/tex].
Therefore, the formula becomes:
[tex]\[
S_8 = \frac{5(1 - \left(\frac{4}{3}\right)^8)}{-\frac{1}{3}}
\][/tex]
Simplify the equation further:
[tex]\[
S_8 = 5 \cdot 3 \cdot (1 - \left(\frac{4}{3}\right)^8)
\][/tex]
[tex]\[
S_8 = 15 \cdot (1 - \left(\frac{4}{3}\right)^8)
\][/tex]
After performing the calculations:
[tex]\[
S_8 \approx 134.83
\][/tex]
Thus, the correct answer is:
B. [tex]\(\quad 134.83\)[/tex]
