To determine the correct formula for this sequence, we observe the given terms: [tex]\( -81, 108, -144, 192, \ldots \)[/tex].
1. Identify the first term ([tex]\( a \)[/tex]):
The first term is [tex]\( -81 \)[/tex].
2. Calculate the common ratio ([tex]\( r \)[/tex]):
To find the common ratio [tex]\( r \)[/tex], we divide the second term by the first term:
[tex]\[
r = \frac{108}{-81} = -\frac{108}{81} = -\frac{4}{3}
\][/tex]
3. Verify consistency of the common ratio:
- For the second and third terms:
[tex]\[
\frac{-144}{108} = -\frac{144}{108} = -\frac{4}{3}
\][/tex]
- For the third and fourth terms:
[tex]\[
\frac{192}{-144} = -\frac{192}{144} = -\frac{4}{3}
\][/tex]
Since the common ratio [tex]\( r \)[/tex] is consistent for all pairs of subsequent terms, we confirm that [tex]\( r = -\frac{4}{3} \)[/tex].
4. Formulate the general term of the sequence:
The general term of a geometric sequence can be described by:
[tex]\[
f(x) = a \cdot r^{x-1}
\][/tex]
Where [tex]\( a \)[/tex] is the first term and [tex]\( r \)[/tex] is the common ratio. Substituting the values we found:
[tex]\[
f(x) = -81 \left(-\frac{4}{3}\right)^{x-1}
\][/tex]
Thus, the formula that can be used to describe the sequence [tex]\( -81, 108, -144, 192, \ldots \)[/tex] is:
[tex]\[
f(x) = -81 \left(-\frac{4}{3}\right)^{x-1}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{f(x)=-81\left(-\frac{4}{3}\right)^{x-1}}
\][/tex]
