To find the function that represents the geometric sequence 6, 18, 54, ..., we need to identify the first term and the common ratio.
- First term, a: The first term of the sequence is 6.
- Common ratio, r: To find the common ratio, we divide the second term by the first term: \[ r = \frac{18}{6} = 3 \]
The formula for the nth term of a geometric sequence can be expressed as: \[ f(n) = a \cdot r^{(n-1)} \] where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
In this case:
- \(a = 6\)
- \(r = 3\)
So, the function representing the geometric sequence is: \[ f(n) = 6 \cdot 3^{(n-1)} \]
Looking at the options provided:
- f(x) = 3(6)^x is incorrect because it does not match the pattern of the sequence.
- f(x) = 3(6)^(x−1) is also incorrect, as it changes the base.
- f(x) = 6(3)^x is the correct representation but in a different form than expected. If we let \(x\) represent \(n-1\), then:
Correct answer: \(f(x) = 6(3)^{(x-1)}\) is not provided directly but adjusts if \(x\) is the term number.
Best match: \(f(x) = 6(3)^x\) interpreted correctly for this context (if \(x\) is shifted appropriately for \(n\)).
Therefore, you may want to select the option that corresponds to your understanding based on how the problem is framed. In most contexts, it would be:
Final choice: \(f(x) = 6(3)^x\) where x would be adjusted based on the term you are looking for (but verify how 'x' is treated).
1 answer