To determine which of the given sequences is a geometric sequence, let's analyze each option step by step.
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
### Option A: [tex]\( 2, -3, \frac{9}{2}, -\frac{18}{4} \)[/tex]
1. To check if this sequence is geometric, we should identify the common ratio by dividing each term by the preceding term.
2. Find the ratio between the second term and the first term:
[tex]\[
\frac{-3}{2} = -1.5
\][/tex]
3. Find the ratio between the third term and the second term:
[tex]\[
\frac{\frac{9}{2}}{-3} = \frac{9}{2} \times \frac{1}{-3} = -1.5
\][/tex]
4. Find the ratio between the fourth term and the third term:
[tex]\[
\frac{-\frac{18}{4}}{\frac{9}{2}} = \frac{-18}{4} \times \frac{2}{9} = -1
\][/tex]
Since the last ratio [tex]\(-1\)[/tex] does not match the previous ratios [tex]\(-1.5\)[/tex], this is not a geometric sequence.
### Option B: [tex]\( -7, 10, 23, 36, \ldots \)[/tex]
1. Check the ratio between the second term and the first term:
[tex]\[
\frac{10}{-7} \approx -1.42857
\][/tex]
2. Check the ratio between the third term and the second term:
[tex]\[
\frac{23}{10} = 2.3
\][/tex]
3. Check the ratio between the fourth term and the third term:
[tex]\[
\frac{36}{23} \approx 1.565
\][/tex]
The ratios [tex]\(-1.42857\)[/tex], [tex]\(2.3\)[/tex], and [tex]\(1.565\)[/tex] are not constant. Thus, this is not a geometric sequence.
### Option C: [tex]\( 0, 1, 2, 3, \ldots \)[/tex]
1. Check the ratio between the second term and the first term:
[tex]\[
\frac{1}{0} \quad \text{(Undefined)}
\][/tex]
Since the first term is [tex]\(0\)[/tex], and division by zero is undefined, this sequence cannot have a common ratio. Hence, this is not a geometric sequence.
### Option D: [tex]\( 8, 4, 2, 1, \frac{1}{2}, \frac{1}{4}, \ldots \)[/tex]
1. Check the ratio between the second term and the first term:
[tex]\[
\frac{4}{8} = \frac{1}{2}
\][/tex]
2. Check the ratio between the third term and the second term:
[tex]\[
\frac{2}{4} = \frac{1}{2}
\][/tex]
3. Check the ratio between the fourth term and the third term:
[tex]\[
\frac{1}{2} = \frac{1}{2}
\][/tex]
4. Check the ratio between the fifth term and the fourth term:
[tex]\[
\frac{\frac{1}{2}}{1} = \frac{1}{2}
\][/tex]
5. Check the ratio between the sixth term and the fifth term:
[tex]\[
\frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{4} \times 2 = \frac{1}{2}
\][/tex]
Since each ratio is [tex]\(\frac{1}{2}\)[/tex] and is consistent throughout, this sequence is a geometric sequence with a common ratio of [tex]\( \frac{1}{2} \)[/tex].
Therefore, the correct answer is:
Option D: [tex]\( 8, 4, 2, 1, \frac{1}{2}, \frac{1}{4}, \ldots \)[/tex].
