Given the sequence:

[tex]\[ -81, 108, -144, 192, \ldots \][/tex]

Which formula can be used to describe the sequence?

A. [tex]\( f(x) = -81 \left(\frac{4}{3}\right)^{x-1} \)[/tex]
B. [tex]\( f(x) = -81 \left(-\frac{3}{4}\right)^{x-1} \)[/tex]
C. [tex]\( f(x) = -81 \left(-\frac{4}{3}\right)^{x-1} \)[/tex]
D. [tex]\( f(x) = -81 \left(\frac{3}{4}\right)^{x-1} \)[/tex]



Answer :

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]