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 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**.