Question

Write a function to represent the geometric sequence 6, 18, 54, . . . .
A. f (x) = 6(3)x
B. f (x) = 3(6)x-1
C. f(x) = 6(3)x-1
D. f(x) = 3(6)x

Answer 1

To represent the geometric sequence 6, 18, 54, ..., we first identify the common ratio and the initial term.

1. The first term \( a = 6 \).
2. The common ratio \( r \) can be determined by dividing the second term by the first term:
\[
r = \frac{18}{6} = 3
\]

The general form of a geometric sequence can be expressed as:
\[
f(x) = a \cdot r^{(x-1)}
\]
where \( a \) is the first term and \( r \) is the common ratio.

For this sequence:
- \( a = 6 \)
- \( r = 3 \)

Thus, we can write the function as:
\[
f(x) = 6 \cdot 3^{(x-1)}
\]

Now we can match this with the provided options:
- A. \( f (x) = 6(3)x \)
- B. \( f (x) = 3(6)x-1 \)
- C. \( f(x) = 6(3)x-1 \)
- D. \( f(x) = 3(6)x \)

The correct representation, based on our derived function, is:
**C. \( f(x) = 6(3)^{x-1} \)**.