- A geometric sequence is defined by the relation \(a_{n+1} = a_n \cdot r\).
- The given rule states each term is the previous term divided by 2, which is equivalent to multiplying by \(\frac{1}{2}\).
- Equating \(a_{n+1} = a_n \cdot r\) and \(a_{n+1} = a_n \cdot \frac{1}{2}\) yields the common ratio \(r = \frac{1}{2}\).
- The correct common ratio is \(\boxed{\frac{1}{2}}\).
Explanation
- Understanding Geometric Sequences
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio, denoted by \(r\). We are given a sequence that starts with \(a_1 = 12\) and follows the rule that each number is the previous number divided by \(2\). Mathematically, this rule is expressed as \(a_{n+1} = \frac{a_n}{2}\).
- Determining the Common Ratio
The standard recurrence relation for a geometric sequence is \(a_{n+1} = a_n \cdot r\). By comparing this definition with the given rule \(a_{n+1} = \frac{a_n}{2}\), we can see that:
\[a_n \cdot r = \frac{a_n}{2}\]
Dividing both sides by \(a_n\) (since the sequence terms are non-zero), we obtain:
\[r = \frac{1}{2}\]
Thus, the common ratio of the geometric sequence is \(\frac{1}{2}\).
- Conclusion and Final Answer
We have determined that the common ratio \(r\) is \(\frac{1}{2}\). Comparing this to the given options:
The correct choice is \(\frac{1}{2}\). Therefore, the common ratio that can be used in an explicit formula to represent the sequence is \(\boxed{\frac{1}{2}}\).
Examples
Imagine you are cutting a piece of paper in half repeatedly. If the original paper has an area of 12 square units, the sequence of areas of the paper pieces would be 12, 6, 3, 1.5, and so on. This is a geometric sequence where each step involves multiplying the previous area by a ratio of \(r = \frac{1}{2}\). Recognizing this ratio helps in modeling various processes like population decay, radioactive half-life, or recursive resource division.