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 ration in an explicit formula that represents the sequence
The Correct Answer and Explanation is:
Correct Answer:
The value that can be used as the common ratio in an explicit formula for this sequence is: 12\boxed{\frac{1}{2}}21
Explanation
A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous one by a constant called the common ratio (usually denoted by rrr).
In this problem, the sequence begins with 12 and each subsequent term is divided by 2. Mathematically, dividing by 2 is the same as multiplying by 12\frac{1}{2}21. So the sequence looks like this: 12, 6, 3, 1.5, 0.75, …12,\ 6,\ 3,\ 1.5,\ 0.75,\ \dots12, 6, 3, 1.5, 0.75, …
To check this, we can divide each term by the previous term:
- 612=12\frac{6}{12} = \frac{1}{2}126=21
- 36=12\frac{3}{6} = \frac{1}{2}63=21
- 1.53=12\frac{1.5}{3} = \frac{1}{2}31.5=21
This confirms that the common ratio r=12r = \frac{1}{2}r=21.
Explicit Formula
An explicit formula for a geometric sequence is: an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}an=a1⋅rn−1
Where:
- ana_nan is the nnn-th term,
- a1a_1a1 is the first term of the sequence,
- rrr is the common ratio,
- nnn is the term number.
In this case:
- a1=12a_1 = 12a1=12,
- r=12r = \frac{1}{2}r=21,
So the explicit formula becomes: an=12⋅(12)n−1a_n = 12 \cdot \left(\frac{1}{2}\right)^{n-1}an=12⋅(21)n−1
This formula allows us to find any term in the sequence directly, without computing all the previous terms.
Conclusion
The common ratio in this geometric sequence is 12\boxed{\frac{1}{2}}21, since each term is found by multiplying the previous term by 12\frac{1}{2}21. This common ratio is essential in writing the explicit formula, which models the pattern of the sequence precisely.
