To determine which of the given options correctly evaluates the series [tex]\(\sum_{k=1}^9 2(4)^{k-1}\)[/tex], we need to recognize that this series represents a geometric series. A geometric series is a series where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio [tex]\(r\)[/tex].
The standard formula to find the sum [tex]\(S_n\)[/tex] of the first [tex]\(n\)[/tex] terms of a geometric series is:
[tex]\[ S_n = a \frac{r^n - 1}{r - 1} \][/tex]
where:
- [tex]\(a\)[/tex] is the first term,
- [tex]\(r\)[/tex] is the common ratio,
- [tex]\(n\)[/tex] is the number of terms.
In our series [tex]\(\sum_{k=1}^9 2(4)^{k-1}\)[/tex]:
- The first term [tex]\(a\)[/tex] is [tex]\(2\)[/tex].
- The common ratio [tex]\(r\)[/tex] is [tex]\(4\)[/tex].
- The number of terms [tex]\(n\)[/tex] is [tex]\(9\)[/tex].
Plugging these values into the formula gives us:
[tex]\[ S_9 = 2 \frac{4^9 - 1}{4 - 1} \][/tex]
To match this with the options, we should rewrite it slightly:
[tex]\[ S_9 = 2 \left( \frac{4^9 - 1}{3} \right) \][/tex]
Among the given options:
1. [tex]\( 2 \left( \frac{1 - 4^8}{1 - 4} \right) \)[/tex]
2. [tex]\( 2 \left( \frac{1 - 4^{10}}{1 - 4} \right) \)[/tex]
3. [tex]\( 2 \left( \frac{1 - 4^9}{1 - 4} \right) \)[/tex]
4. [tex]\( 2 \left( \frac{(1 - 4)^9}{1 - 4} \right) \)[/tex]
We need to find the option that matches [tex]\( 2 \left( \frac{4^9 - 1}{3} \right) \)[/tex]. Simplifying, we see that:
[tex]\[ 2 \left( \frac{1 - 4^9}{1 - 4} \right) = 2 \left( \frac{-4^9 + 1}{-3} \right) = 2 \left( \frac{4^9 - 1}{3} \right) \][/tex]
So, the correct option is:
[tex]\[ 2 \left( \frac{1 - 4^9}{1 - 4} \right) \][/tex]
Thus, the correct option to evaluate the series [tex]\(\sum_{k=1}^9 2(4)^{k-1}\)[/tex] is:
[tex]\[ \boxed{2 \left( \frac{1 - 4^9}{1 - 4} \right)} \][/tex]
