To represent the geometric sequence 6, 18, 54, ..., we first need to identify the first term and the common ratio:
- The first term \( a = 6 \)
- The common ratio \( r = \frac{18}{6} = 3 \)
A geometric sequence can be described by the formula:
\[ f(n) = a \cdot r^{(n-1)} \]
Where \( n \) is the term number (starting from 1).
In this case:
\[ f(n) = 6 \cdot 3^{(n-1)} \]
Now, we will analyze the given options to find an appropriate function. We can adjust the sequence functionally to represent it with \( x \) instead of \( n \) (where \( x \) corresponds to \( n \)).
So we have:
- For \( n = 1 \): \( f(1) = 6 \cdot 3^{0} = 6 \)
- For \( n = 2 \): \( f(2) = 6 \cdot 3^{1} = 18 \)
- For \( n = 3 \): \( f(3) = 6 \cdot 3^{2} = 54 \)
Thus, the correct formulation in terms of \( x \) is:
\[ f(x) = 6 \cdot 3^{(x-1)} \]
This matches with option C:
C) \( f(x) = 6(3)^{x-1} \)
So the correct answer is C.
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