To determine which formula describes the given sequence [tex]\(-\frac{2}{3}, -4, -24, -144, \ldots\)[/tex], we need to recognize that this is a geometric sequence. A geometric sequence has the form:
[tex]\[ a_n = a_1 \cdot r^{n-1} \][/tex]
where [tex]\(a_1\)[/tex] is the first term, [tex]\(r\)[/tex] is the common ratio, and [tex]\(n\)[/tex] is the term number.
Step 1: Identifying the First Term ([tex]\(a_1\)[/tex])
The first term in the sequence is given as [tex]\( a_1 = -\frac{2}{3} \)[/tex].
Step 2: Identifying the Second Term ([tex]\(a_2\)[/tex])
The second term in the sequence is given as [tex]\( a_2 = -4 \)[/tex].
Step 3: Calculating the Common Ratio ([tex]\(r\)[/tex])
The common ratio [tex]\( r \)[/tex] is the factor by which we multiply to get from one term to the next. It can be found by dividing the second term by the first term:
[tex]\[ r = \frac{a_2}{a_1} = \frac{-4}{-\frac{2}{3}} = -4 \cdot \left(-\frac{3}{2}\right) = 6 \][/tex]
So, [tex]\( r = 6 \)[/tex].
Step 4: Writing the General Formula
The general formula for the [tex]\(n\)[/tex]-th term of a geometric sequence is:
[tex]\[ a_n = a_1 \cdot r^{n-1} \][/tex]
Substituting the values of [tex]\(a_1\)[/tex] and [tex]\(r\)[/tex]:
[tex]\[ a_n = -\frac{2}{3} \cdot 6^{n-1} \][/tex]
Step 5: Matching the Given Formulas
We need to match the correct formula from the provided options. Which means, the formula that describes our geometric sequence is:
[tex]\[ f(x) = -\frac{2}{3}(6)^{x-1} \][/tex]
Thus, the choice that corresponds to the sequence [tex]\(-\frac{2}{3}, -4, -24, -144, \ldots\)[/tex] is:
[tex]\[ f(x) = -\frac{2}{3}(6)^{x-1} \][/tex]
