A sequence of numbers begins with 12 and progresses geometrically. Each number is the previous number divided by 2.
Which value can be used as the common ratio in an explicit formula that represents the sequence?
2
6
12
The Correct Answer and Explanation is:
In a geometric sequence, each term is obtained by multiplying the previous term by a constant value, known as the common ratio (denoted as r). The general form of an explicit formula for the n-th term of a geometric sequence is:
[
a_n = a_1 \cdot r^{n-1}
]
Where:
- ( a_n ) is the n-th term of the sequence,
- ( a_1 ) is the first term of the sequence,
- ( r ) is the common ratio, and
- ( n ) is the position of the term in the sequence.
Given in the problem, the sequence starts with 12, and each number is the previous number divided by 2. This means that to get from one term to the next, you are dividing by 2, or equivalently, multiplying by ( \frac{1}{2} ).
Step-by-step analysis:
- First term (a₁): The first term in the sequence is 12, which is given. Thus, ( a_1 = 12 ).
- Finding the common ratio (r): According to the problem, each term is the previous term divided by 2. Therefore, the common ratio ( r ) is:
[
r = \frac{1}{2}
] - Example of the sequence progression:
- ( a_1 = 12 )
- ( a_2 = 12 \div 2 = 6 )
- ( a_3 = 6 \div 2 = 3 )
- ( a_4 = 3 \div 2 = 1.5 )
- And so on.
This pattern confirms that each term is indeed being divided by 2, which corresponds to multiplying by ( \frac{1}{2} ).
Conclusion:
The common ratio ( r ) for this geometric sequence is ( \frac{1}{2} ). Among the options provided (2, 6, 12), none of them exactly matches the correct ratio of ( \frac{1}{2} ), which means that none of the given options is correct. However, the correct common ratio is ( \frac{1}{2} ). The problem might be asking for a number that represents the change in value (i.e., division by 2), so the correct value, expressed in another way, would be ( \frac{1}{2} ), but since that is not listed as an option, clarification may be needed.