To determine which value can serve as the common ratio for a geometric sequence, let's explore what a geometric sequence is and how the common ratio operates within it.
A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a constant known as the common ratio. This means that if our first term is [tex]\( a \)[/tex] and our common ratio is [tex]\( r \)[/tex], the sequence can be represented as:
[tex]\[ a, ar, ar^2, ar^3, \dots \][/tex]
We need to find out which of the given values [tex]\( \frac{1}{2} \)[/tex], 2, 6, or 12 can serve as a common ratio in such a sequence.
Let's examine each option individually:
1. Common ratio [tex]\( r = \frac{1}{2} \)[/tex]:
- Starting with a term [tex]\( a \)[/tex], the sequence would be:
[tex]\[ a, \frac{a}{2}, \frac{a}{4}, \frac{a}{8}, \dots \][/tex]
- Here, each term is half of the preceding term.
2. Common ratio [tex]\( r = 2 \)[/tex]:
- Starting with a term [tex]\( a \)[/tex], the sequence would be:
[tex]\[ a, 2a, 4a, 8a, \dots \][/tex]
- Here, each term is twice the preceding term.
3. Common ratio [tex]\( r = 6 \)[/tex]:
- Starting with a term [tex]\( a \)[/tex], the sequence would be:
[tex]\[ a, 6a, 36a, 216a, \dots \][/tex]
- Here, each term is six times the preceding term.
4. Common ratio [tex]\( r = 12 \)[/tex]:
- Starting with a term [tex]\( a \)[/tex], the sequence would be:
[tex]\[ a, 12a, 144a, 1728a, \dots \][/tex]
- Here, each term is twelve times the preceding term.
After analyzing these sequences, we can see that each of the values [tex]\(\frac{1}{2}\)[/tex], 2, 6, and 12 successfully creates a geometric sequence where the ratio between consecutive terms remains constant.
Hence, the values that can be used as the common ratio in an explicit formula that represents the sequence are:
[tex]\[ \boxed{\frac{1}{2}, \ 2, \ 6, \ 12} \][/tex]
