To determine the correct explicit function for the given geometric sequence [tex]\(3, 1, \frac{1}{3}, \frac{1}{9}, \ldots\)[/tex], we need to use the general form of the [tex]\(n\)[/tex]-th term of a geometric sequence, which is given by:
[tex]\[ f(n) = a \cdot r^{(n-1)} \][/tex]
where:
- [tex]\(a\)[/tex] is the first term of the sequence.
- [tex]\(r\)[/tex] is the common ratio between the consecutive terms.
Let's identify the first term ([tex]\(a\)[/tex]) and the common ratio ([tex]\(r\)[/tex]) from the given sequence:
- The first term [tex]\(a\)[/tex] is [tex]\(3\)[/tex].
- The common ratio [tex]\(r\)[/tex] is calculated by dividing the second term by the first term, i.e., [tex]\(r = \frac{1}{3 \div 3} = \frac{1}{3}\)[/tex].
Using these values, the general form of the [tex]\(n\)[/tex]-th term for this sequence is:
[tex]\[ f(n) = 3 \cdot \left(\frac{1}{3}\right)^{(n-1)} \][/tex]
Now we will verify this function against each of the answer options:
A. [tex]\(f(n) = \frac{1}{3} \cdot 2^{(n-1)}\)[/tex]
- This does not match our derived form because the common ratio should be [tex]\(\frac{1}{3}\)[/tex] not [tex]\(2\)[/tex].
B. [tex]\(f(n) = 3 \cdot \left(\frac{1}{3}\right)^{(n-1)}\)[/tex]
- This matches our derived form exactly.
C. [tex]\(f(n) = 2 \cdot \left(\frac{1}{3}\right)^{(n-1)}\)[/tex]
- This does not match our derived form because the first term should be [tex]\(3\)[/tex], not [tex]\(2\)[/tex].
D. [tex]\(f(n) = \frac{1}{3} \cdot 3^{(n-1)}\)[/tex]
- This is incorrect because it would produce a different sequence where the common ratio's exponent is positive, not negative.
Thus, the correct answer is:
B. [tex]\(f(n)=3 \cdot\left(\frac{1}{3}\right)^{(n-1)}\)[/tex]
Therefore, "B is the correct answer".
