To determine the explicit formula for the given geometric sequence [tex]\(\{12, -6, 3, \ldots\}\)[/tex], let's break it down step-by-step.
### Given Sequence
The given sequence is: [tex]\(12, -6, 3, \ldots\)[/tex]
This is a geometric sequence, so we need to identify the first term ([tex]\(a_1\)[/tex]) and the common ratio ([tex]\(r\)[/tex]).
### Identifying the First Term and Common Ratio
1. The first term [tex]\(a_1\)[/tex] is the first element of the sequence, which is [tex]\(12\)[/tex].
2. The common ratio [tex]\(r\)[/tex] is found by dividing any term by the preceding term:
- The second term is [tex]\(-6\)[/tex], and the first term is [tex]\(12\)[/tex], so:
[tex]\[
r = \frac{-6}{12} = -\frac{1}{2}
\][/tex]
- To confirm, the third term is [tex]\(3\)[/tex], and the second term is [tex]\(-6\)[/tex], so:
[tex]\[
r = \frac{3}{-6} = -\frac{1}{2}
\][/tex]
Thus, [tex]\(a_1 = 12\)[/tex] and [tex]\(r = -\frac{1}{2}\)[/tex].
### General Formula for a Geometric Sequence
The explicit formula for the [tex]\(n\)[/tex]-th term of a geometric sequence is given by:
[tex]\[
a_n = a_1 \cdot r^{n-1}
\][/tex]
### Substituting Values
Using [tex]\(a_1 = 12\)[/tex] and [tex]\(r = -\frac{1}{2}\)[/tex], the formula becomes:
[tex]\[
a_n = 12 \cdot \left(-\frac{1}{2}\right)^{n-1}
\][/tex]
### Checking Given Formulas
Now, we need to check the provided formulas to determine which one matches our derived formula:
1. [tex]\(a_n = 12 \cdot \left(-\frac{1}{2}\right)^{n-1}\)[/tex]
[tex]\[
a_1 = 12 \cdot \left(-\frac{1}{2}\right)^{1-1} = 12 \cdot (1) = 12
\][/tex]
The first term matches, so this formula is correct.
2. [tex]\(a_n = \frac{1}{2} \cdot (-12)^{n-1}\)[/tex]
[tex]\[
a_1 = \frac{1}{2} \cdot (-12)^{1-1} = \frac{1}{2} \cdot (1) = \frac{1}{2}
\][/tex]
This value does not match the first term [tex]\(a_1 = 12\)[/tex].
3. [tex]\(a_n = -12 \cdot \frac{1}{2}^{n-1}\)[/tex]
[tex]\[
a_1 = -12 \cdot \frac{1}{2}^{1-1} = -12 \cdot (1) = -12
\][/tex]
This value does not match the first term [tex]\(a_1 = 12\)[/tex].
4. [tex]\(a_n = -\frac{1}{2} \cdot (12)^{n-1}\)[/tex]
[tex]\[
a_1 = -\frac{1}{2} \cdot (12)^{1-1} = -\frac{1}{2} \cdot (1) = -\frac{1}{2}
\][/tex]
This value does not match the first term [tex]\(a_1 = 12\)[/tex].
### Conclusion
The correct explicit formula for the given geometric sequence [tex]\(\{12, -6, 3, \ldots\}\)[/tex] is:
\[
a_n = 12 \cdot \left(-\frac{1}{2}\right)^{n-1}
\
