In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio. Given that each number in our sequence is the previous number divided by 2, let's determine the common ratio.
1. Start by understanding what it means to divide by 2 in the context of a geometric sequence. Dividing by 2 can be written as multiplying by [tex]\(\frac{1}{2}\)[/tex]. Let's consider the first few terms of the sequence to see how this works.
2. The first term of the sequence is 12.
[tex]\[
a_1 = 12
\][/tex]
3. To find the second term, divide the first term by 2:
[tex]\[
a_2 = 12 \div 2 = 12 \times \frac{1}{2} = 6
\][/tex]
4. To find the third term, divide the second term by 2:
[tex]\[
a_3 = 6 \div 2 = 6 \times \frac{1}{2} = 3
\][/tex]
5. To find the fourth term, divide the third term by 2:
[tex]\[
a_4 = 3 \div 2 = 3 \times \frac{1}{2} = 1.5
\][/tex]
6. From these calculations, you can see a pattern: each term is being multiplied by [tex]\(\frac{1}{2}\)[/tex]. This consistent factor is the common ratio.
Therefore, the common ratio [tex]\(r\)[/tex] that can be used in the explicit formula for this geometric sequence is:
[tex]\[
r = \frac{1}{2}
\][/tex]
So, the value that can be used as the common ratio in the explicit formula representing this sequence is [tex]\(\frac{1}{2}\)[/tex].
