Question

Write a function to represent the geometric sequence 6, 18, 54, . . . .(1 point)
Responses

f(x)=6(3)x−1

(

)
=
6
(
3
)

−
1
f left parenthesis x right parenthesis equals 6 left parenthesis 3 right parenthesis superscript x minus 1 baseline

f(x)=3(6)x−1

(

)
=
3
(
6
)

−
1
f left parenthesis x right parenthesis equals 3 left parenthesis 6 right parenthesis superscript x minus 1 baseline

f(x)=6(3)x

(

)
=
6
(
3
)

f left parenthesis x right parenthesis equals 6 left parenthesis 3 right parenthesis superscript x baseline

f(x)=3(6)x

Answer 1

To represent the geometric sequence \(6, 18, 54, \ldots\), we first recognize the pattern in the sequence. The first term \(a_1\) is 6, and each subsequent term is obtained by multiplying the previous term by the common ratio, which is 3.

We can express the \(n\)-th term of a geometric sequence with the formula:
\[
a_n = a_1 \cdot r^{(n-1)}
\]
where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

In this case:
- \(a_1 = 6\)
- \(r = 3\)

Thus, the formula for the \(n\)-th term is:
\[
a_n = 6 \cdot 3^{(n-1)}
\]

If we define a function \(f(x)\) to represent the \(n\)-th term of the sequence, we can write:
\[
f(x) = 6 \cdot 3^{(x-1)}
\]

So the correct representation of the sequence as a function is:
\[
f(x) = 6(3)^{(x-1)}
\]

This matches the first option provided:

**Response:** \(f(x) = 6(3)^{x-1}\)