To determine which of the given sequences is a geometric sequence, we need to check if there is a constant ratio between successive terms.
A sequence is geometric if the ratio between consecutive terms is constant. Formulaically, a sequence [tex]\( a_1, a_2, a_3, \ldots \)[/tex] is geometric if there exists a constant ratio [tex]\( r \)[/tex] such that:
[tex]\[ \frac{a_{n+1}}{a_n} = r \quad \text{for all } n \][/tex]
Now, let's examine each sequence:
1. Sequence: [tex]\( 3, 12, 48, \ldots \)[/tex]
- Compute the ratios:
[tex]\[
\frac{12}{3} = 4, \quad \frac{48}{12} = 4
\][/tex]
- Since both ratios are equal to 4, the ratio is constant.
- Therefore, [tex]\( 3, 12, 48, \ldots \)[/tex] is a geometric sequence with a common ratio of 4.
2. Sequence: [tex]\( 3, 1, \frac{1}{2}, \ldots \)[/tex]
- Compute the ratios:
[tex]\[
\frac{1}{3} = \frac{1}{3}, \quad \frac{\frac{1}{2}}{1} = \frac{1}{2}
\][/tex]
- The ratios [tex]\( \frac{1}{3} \)[/tex] and [tex]\( \frac{1}{2} \)[/tex] are not the same.
- Therefore, [tex]\( 3, 1, \frac{1}{2}, \ldots \)[/tex] is not a geometric sequence.
3. Sequence: [tex]\( 0, 1, 3, 9, \ldots \)[/tex]
- Compute the first ratio:
[tex]\[
\frac{1}{0}
\][/tex]
- The ratio is undefined because division by zero is undefined.
- Therefore, [tex]\( 0, 1, 3, 9, \ldots \)[/tex] is not a geometric sequence.
4. Sequence: [tex]\( 3, 6, 12, 21, \ldots \)[/tex]
- Compute the ratios:
[tex]\[
\frac{6}{3} = 2, \quad \frac{12}{6} = 2, \quad \frac{21}{12} = 1.75
\][/tex]
- The ratios [tex]\( 2 \)[/tex] and [tex]\( 1.75 \)[/tex] are not the same.
- Therefore, [tex]\( 3, 6, 12, 21, \ldots \)[/tex] is not a geometric sequence.
Conclusion:
Among the given sequences, only the sequence [tex]\( 3, 12, 48, \ldots \)[/tex] is a geometric sequence. It has a common ratio of 4.
