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Which of the following can be used to evaluate the series [tex]\(\sum_{k=1}^9 2(4)^{k-1}\)[/tex]?

A. [tex]\(2\left(\frac{1-4^8}{1-4}\right)\)[/tex]

B. [tex]\(2\left(\frac{1-4^{10}}{1-4}\right)\)[/tex]

C. [tex]\(2\left(\frac{1-4^9}{1-4}\right)\)[/tex]

D. [tex]\(2\left(\frac{(1-4)^9}{1-4}\right)\)[/tex]


Sagot :

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]